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STATIONARY AND 2+1 DIMENSIONALINTEGRABLE REDUCTIONS OF AKNS HIERARCHY
MELTEM LEMAN YERBAG FRANCISCO
January, 2004
Stationary and 2+1 Dimensional IntegrableReductions of AKNS Hierarchy
By
MELTEM LEMAN YERBAG FRANCISCO
A Dissertation Submitted to theGraduate School in Partial Fulfillment of the
Requirements for the Degree of
MASTER OF SCIENCE
Department: MathematicsMajor : Mathematics
Izmir Institute of Technology
Izmir, Turkey
January, 2004
We approve the thesis of the Meltem Leman Yerbag Francisco
Date of Signature
—————————————————— 15.01.2004
Prof. Dr. Oktay PASHAEV
Supervisor
Department of Mathematics
—————————————————— 15.01.2004
Prof. Dr. Recai ERDEM
Department of Physics
—————————————————— 15.01.2004
Prof. Dr. Acar SAVACI
Department of Electrical Engineering
—————————————————— 15.01.2004
Yrd. Doc. Dr. Gamze TANOGLU
Head of the Department of Mathematics
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ACKNOWLEDGEMENTS
I would like to give my endless gratitude to Prof.Dr.Oktay Pashaev for sug-gesting the problems, giving me the much needed inspiration from beginning toend and being for me the best and most patient supervisor during the preparationof this thesis.
I am so grateful to Prof.Dr.Allan Fordy for his valuable suggestions, dis-cussions and sending the paper about Henon-Heiles system.
I am indebted to my friend Fatih Erman for the contributions he made inpreparing the final LATEX form of this thesis.
Special thanks go to the department of mathematics for hospitality andthe research assistants in my office for their help and friendship.
Finally I would like to thank to my husband and my parents for theirsupport, help and patience.
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ABSTRACT
The main concepts of the soliton theory and infinite dimensional Hamil-tonian Systems, including AKNS (Ablowitz, Kaup, Newell, Segur) integrablehierarchy of nonlinear evolution equations are introduced. By integro-differentialrecursion operator for this hierarchy, several reductions to KDV, MKdV, mixedKdV/MKdV and Reaction-Diffusion system are constructed. The stationary re-duction of the fifth order KdV is related to finite-dimensional integrable systemof Henon-Heiles type. Different integrable extensions of Henon-Heiles model arefound with corresponding separation of variables in Hamilton-Jacobi theory. Us-ing the second and the third members of AKNS hierarchy, new method to solve2+1 dimensional Kadomtsev-Petviashvili(KP-II) equation is proposed. By theHirota bilinear method, one and two soliton solutions of KP-II are constructedand the resonance character of their mutual interactions are studied. By ourbilinear form we first time created new four virtual soliton resonance solution forKPII. Finally, relations of our two soliton solution with degenerate four solitonsolution in canonical Hirota form of KPII are established.
iii
OZET
Soliton teorisinin ana kavramları ve lineer olmayan evrim denklemlerininAKNS (Ablowitz, Kaup, Newel, Segur) integrallenebilir hiyerarsisini iceren son-suz boyutlu hamiltoniyen sistemlerine bir giris yapıldı. Bu hiyerarside cesitli in-dirgemeler yapılarak, integro-differensiyel tekrarlama operatoru yardımı ile, KdV,MKdV, ve karısık KdV/MKdV nin ve de reaksiyon-difuzyon denklemleri eldeedildi. Besinci derece KdV nin duragan indirgenmesi, Henon-Heiles tipi sonluboyutlu integrallenebilir sistemi ile iliskilidir. Henon-Heiles tipi sonlu boyutlu in-tegrallenebilir uzantilari, Hamilton-Jacobi teorisindeki ilgili degiskenlerin yardımıile bulundu. AKNS hiyerarsisinin ikinci ve ucuncu uyelerini kullanarak 2+1boyutlu Kadomtsev-Petviashivili (KPII) denklemini cozmek icin yeni bir yontemsunuldu. Hirota bilineer yontemi vasıtasıyla KPII nin bir ve iki soliton cozumleribulundu ve ayrıca bunların karsılıklı etkilesimlerinin rezonans karakteri calısıldı.Yeni buldugumuz bilineer form ile ilk defa KP icin dort sanal soliton rezonanscozumunu elde ettik. Son olarak Satsuma ve Hirotanın yozlasmıs dort solitoncozumu ile bizim iki soliton cozumumuz arasındaki iliski kuruldu.
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TABLE OF CONTENTS
Chapter 1. INTRODUCTION 21.1 Basic idea of Soliton Theory . . . . . . . . . . . . . . . . . . . . . 6
Chapter 2. EVOLUTION EQUATIONS AND HAMILTONIAN SYSTEMS 82.1 Hamiltonian Dynamical Systems . . . . . . . . . . . . . . . . . . . 82.2 Separation of Variables . . . . . . . . . . . . . . . . . . . . . . . . 92.3 Evolution Equations and Dynamical Systems . . . . . . . . . . . . 102.4 Infinite Dimensional Dynamical Systems . . . . . . . . . . . . . . 112.5 KdV as a Hamiltonian System . . . . . . . . . . . . . . . . . . . . 112.6 Spectral Transform for Linear equations . . . . . . . . . . . . . . 122.7 Nonlinear Fourier Transform . . . . . . . . . . . . . . . . . . . . . 132.8 Linearization of Nonlinear Problem . . . . . . . . . . . . . . . . . 13
Chapter 3. AKNS INTEGRABLE HIERARCHY 153.1 Hierarchy of Integrable Evolutions . . . . . . . . . . . . . . . . . . 153.2 AKNS Hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.3 KdV and MKdV reductions . . . . . . . . . . . . . . . . . . . . . 193.4 The mixed KdV-MKdV hierarchy . . . . . . . . . . . . . . . . . . 19
Chapter 4. GENERALIZED HENON-HEILES SYSTEM 214.1 Henon-Heiles system . . . . . . . . . . . . . . . . . . . . . . . . . 214.2 Stationary reductions from Soliton Equations . . . . . . . . . . . 224.3 The Hamilton-Jacobi theory . . . . . . . . . . . . . . . . . . . . . 234.4 Separation of Variables . . . . . . . . . . . . . . . . . . . . . . . . 25
Chapter 5. SEPARATION OF VARIABLES IN H-H SYSTEM 275.1 Bi-Hamiltonian Formulation . . . . . . . . . . . . . . . . . . . . . 285.2 Integrable Extensions . . . . . . . . . . . . . . . . . . . . . . . . . 29
5.2.1 C-Extended Henon-Heiles System . . . . . . . . . . . . . . 295.2.2 D-Extended Henon-Heiles System . . . . . . . . . . . . . . 325.2.3 The Harmonic Extension . . . . . . . . . . . . . . . . . . 335.2.4 The Harmonic C-Extension . . . . . . . . . . . . . . . . . 355.2.5 The Harmonic D-Extension . . . . . . . . . . . . . . . . . 385.2.6 The Harmonic C-D Extension . . . . . . . . . . . . . . . . 40
v
Chapter 6. RESONANCE SOLITONS IN AKNS HIERARCHY 436.1 Hirota Bilinear Method in Soliton Theory . . . . . . . . . . . . . 436.2 Bilinear Representation of Reaction-Diffusion System . . . . . . . 436.3 Resonance Dynamics of Dissipatons . . . . . . . . . . . . . . . . . 456.4 Geometrical Interpretation . . . . . . . . . . . . . . . . . . . . . . 476.5 Dissipative solitons for the Third Flow . . . . . . . . . . . . . . . 48
6.5.1 One Dissipative Soliton Solution . . . . . . . . . . . . . . . 506.5.2 Two Dissipative Soliton Solution . . . . . . . . . . . . . . 506.5.3 The MKdV Reduction . . . . . . . . . . . . . . . . . . . . 526.5.4 The MKdV-KdV Mixed Reduction . . . . . . . . . . . . . 53
Chapter 7. THE KADOMTSEV-PETVIASHVILI MODEL 557.1 2+1 Dimensional Reduction of AKNS . . . . . . . . . . . . . . . . 557.2 Bilinear Representation of KPII and AKNS flows . . . . . . . . . 577.3 Two Soliton Solutions . . . . . . . . . . . . . . . . . . . . . . . . 607.4 Degenerate Four-Soliton Solution . . . . . . . . . . . . . . . . . . 617.5 Resonance Interaction of Planar Solitons . . . . . . . . . . . . . . 66
Chapter 8. CONCLUSIONS 71
Chapter A. HIROTA DERIVATIVES AND ITS PROPERTIES 83
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LIST OF FIGURES
6.1 Resonant dissipaton creation . . . . . . . . . . . . . . . . . . . . . 466.2 Exchange interactions of dissipatons . . . . . . . . . . . . . . . . . 47
7.1 Resonance KP soliton dynamics . . . . . . . . . . . . . . . . . . . 667.2 Resonance KP soliton dynamics . . . . . . . . . . . . . . . . . . . 677.3 Resonance KP soliton dynamics . . . . . . . . . . . . . . . . . . . 677.4 Resonance KP soliton dynamics . . . . . . . . . . . . . . . . . . . 687.5 Resonance KP soliton dynamics . . . . . . . . . . . . . . . . . . . 687.6 One Virtual Soliton Solution . . . . . . . . . . . . . . . . . . . . . 697.7 Solitons in the ocean . . . . . . . . . . . . . . . . . . . . . . . . . 70
1
Chapter 1
INTRODUCTION
In August of 1834, John Scott Russel (1808-1882) was studying the motion of
a small boat in a canal and he observed that when the boat suddenly stopped,
a lump of water formed at the front of the boat, moved forward with constant
speed and shape. He called this phenomenon as the WAVE OF TRANSLATION.
In a number of experiments he determined the shape of a solitary wave to be that
of sech2x function. This was the first recorded observation of a soliton [1]. At
that time there was no equation describing such water waves.
But in 1895 Korteweg and de Vries [2] were studying propagation of waves on
the surface of shallow water and derived the following nonlinear partial differential
equation called the KdV equation,
Ut = Uxxx − 6UUx (1.1)
where Ut = ∂U/∂t and Uxxx = ∂3U/∂x3. They found the solution of this
equation exactly in the same form as derived by J.Scott Russel. Later this is called
soliton by Kruskal and Zabusky [3], in their study of the KdV as a continuum limit
of the Fermi-Pasta-Ulam chain problem. Two years later the method of solution
of the initial-value problem associated with the KdV equation, for solutions,
rapidly vanishing as x → ±∞, was found in terms of the spectral problem for
the one-dimensional Schrodinger operator [4]. As it was shown, solutions of the
KdV equation corresponding to a purely continuous spectrum are similar to wave
packets, and disperse as time goes on. Instead, solutions with a purely discrete
spectrum describe the interaction of N solitons with each other. The method
of solving the ”inverse problem” , i.e. reconstruction of a potential from the
given spectral data had been established by the Russian mathematical-physics
community in the early 1950s. This method applied for solving the KdV equation
is known as the ”inverse-scattering method” [5].
2
It then was found that for the KdV equation infinitely many constants of
the motion exist and are all functionally independent of each other, like in the
definition of Liouville intergability of finite-dimensional systems [6]. This point
was further clarified by showing that the KdV equation can be written in the
Hamiltonian form [7]. Relation between solitons and conservation laws has been
realized by P. Lax in his representation for the soliton equation, associating the
KdV flow with an isospectral change of the linear Schrodinger operator [8]. Later
it was shown that soliton equations include other nonlinear evolution equations
[9], [10] with rich and beautiful mathematical structures of Hamiltonian inte-
grable systems[11]. Several methods were developed to study such equations:
(a) the Hirota bilinear method [12], [13], allowing to construct the N-soliton so-
lutions as an alternative to the spectral method [14], [15], [16]. (b) Backlund
and Darboux transformations [17], [18]. (c) The zero-curvature formulation of
the linear problem [19], [20], [21], [22]. (d) The algebraic-geometric formula-
tion for the class of periodic, finite-gap solutions [23], [24], [25], [26]. (e) The
generalized Wronskian method [15]. (g) The Riemann-Hilbert problem and its
generalization to the so-called DBAR problem [27], [28], [29], [30]. During the
last decades, it has become more widely recognized in many areas of physics [31],
[32], such as hydrodynamics [33], [34], [35], plasma physics [36], solid state and
condensed matter physics [37], [38], [39], [40], [41], [42], biology [38], [43], non-
linear optics [44], [45], [46], general relativity [47], [48] and elementary particle
physics [49], [50], [51] that solitons can result in qualitatively new phenomena
which cannot be constructed by perturbation theory of the linear systems [52],
[53], [54], [55], [56]. In the last 20 years, optical solitons have been discovered
and investigated in different systems like optical fibers [57], [58]. Trans-oceanic
high-rate transmission of information by one soliton in nonlinear herbium-doped
fibers, as well as ultrafast pulse generation by a soliton laser, and a number of
all-optical switching devices, with potential use as integrated components of op-
tical computers, show the influence of quite exotic idea on modern development
in science and technology. Soliton idea has stimulated also development and dis-
coveries in mathematics itself [59], [60], [61], [62]. It becomes interdisciplinary
subject attracting researchers from different fields [63], [64], [65], [66], [67], [68],
and appeared now in some textbooks [69], [70], [71].
In this thesis we study the problem of separation of variables for several
integrable extensions of Henon-Heiles system, relations with AKNS integrable
hierarchy and KP equation, corresponding soliton solutions and their resonance
dynamics.
In the next section we present a brief discussion of the main idea of soliton
3
theory.
In Chapter 2 we review the current approach to the integrable evolution equa-
tions as Hamiltonian dynamical systems. In section 2.1 the Hamilton theory in
the symplectic geometry approach and basic ingredients of the Liouville theorem
for integrability of finite dimensional models are introduced. In section 2.2 we
discuss separation of variables and canonical transformation to the action-angle
variables. Definitions and examples of evolution equations and dynamical systems
are subject of section 2.3. Then, in section 2.4 we show that an evolution equa-
tion can be considered as an infinite dimensional dynamical system. For soliton
equations these dynamical systems are Hamiltonian systems. The Hamiltonian
structure of the KdV equation, with the Faddeev-Zakharov Poisson bracket and
infinite set of integrals of motion in involution are given in section 2.5. The idea
of spectral transform is illustrated for the linear equations in section 2.6, while
for the nonlinear equations in section 2.7. The Lax representation, isospectrality
conditions and the zero-curvature representation of a nonlinear integrable system
are exposed in section 2.8.
In the third chapter we introduce the AKNS hierarchy of evolution equations
and its reductions. We show that every integral of motion generates the Hamil-
tonian flow from infinite hierarchy (section 3.1). The AKNS hierarchy, recursion
operator and first members of the hierarchy are studied in details in section 3.2.
In section 3.3 we obtain two important reductions of AKNS hierarchy to the
KdV, MKdV equations, corresponding recursion operators and reduced hierar-
chies. In section 3.4 we found new reduction to the mixed KdV-MKdV equation,
corresponding recursion operator and generated by it an infinite hierarchy.
Chapter 4 is devoted to the Henon-Heiles model which is the subject of recent
intensive studies in integrability and chaos. Formulation of the Henon-Heiles
model and its integrable cases are given in section 4.1. Then, in the next section,
following results of A. Fordy we show how these integrable cases appear from the
stationary reductions of the fifth order soliton equations, namely, the KdV, the
Sawada-Kotera and the Kaup-Kupershmidt equations. For solving Henon-Heiles
model we use the Hamilton-Jacobi theory and separation of variable technique
from this theory, which we introduce in section 4.4
Separation of variables in the Henon-Heiles model and its integrable exten-
sions are considered in Chapter 5. First, we are discussing an additional integral
of motion, Liouville integrability and separation of variables for the Hamiltonian
characteristic function. Then, in section 5.1 we show that the second integral
can be considered as an additional, second Hamiltonian of the Henon-Heiles
system. This type of systems with two Hamiltonians structure are called the
4
bi-Hamiltonian systems. They are subject of recent studies on algebraic for-
mulation of integrability. Separation of variables in the extended Henon-Heiles
model we present in section 5.2. Extension with the constant C term and corre-
sponding bi-Hamiltonian formulation, in the subsection 5.2.1, extension with the
inverse cubic term of strength D, in subsection 5.2.2, harmonic term extension in
subsection 5.2.3, and mixed harmonic C and harmonic D extensions in subsec-
tions 5.2.4, 5.2.5. Separation of variables and bi-Hamiltonian formulation in the
general extension with harmonic terms and C, D terms, is given in subsection
5.2.6.
In the Chapter 6 we deal with exact one and two soliton solutions of the first
two nonlinear systems from the AKNS hierarchy, and their resonance dynamics.
In section 6.1 we introduce the main ingredients of the Hirota bilinear method
to solve soliton equations. Bilinear representation and one and two dissipative
soliton solutions for the Reaction-Diffusion equations are given in section 6.2.
The resonance character of soliton interactions in this case illustrates section 6.3.
Beautiful geometrical interpretation of equations as a constant curvature surface
in pseudo-Riemannian space we demonstrate in section 6.4. One soliton metric in
this case develops the so called causal singularity, similar to black hole horizons
in the General Relativity Theory. New dissipative solitons for the third flow of
AKNS we construct in section 6.5; one soliton solution in subsection 6.5.1 and
two-soliton solution in subsection 6.5.2. Reductions of bilinear equations and
corresponding soliton solutions to the MKdV equation and mixed KdV-MKdV
equations are found in subsections 6.5.3 and 6.5.4 respectively.
In Chapter 7 we propose a new method to generate solutions of 2+1 dimen-
sional extension of KdV equation, known as the KP equation. In section 7.1 we
show that if one considers a simultaneous solution of the second and the third
flows from the AKNS hierarchy, then the product e+e− satisfies the KPII equa-
tion (Theorem 7.1.1). Using this theorem and results of Chapter 6 we construct
new bilinear representation of KPII equation. Then, by our method we construct
one soliton solution (section 7.2) and two soliton solution (section 7.3.). In sec-
tion 7.4 we compare our two soliton solution of KPII with the one of the known
before bilinear Hirota representation. As a result we find that our two-soliton so-
lution corresponds to the degenerate four soliton solution in the standard Hirota
form. Resonance character of our soliton interactions is studied in section 7.5
In Chapter 8 we discuss main results of this thesis and conclusions. In Ap-
pendix we remained basic formulas of the Hirota bilinear method explored in our
work.
5
1.1 Basic idea of Soliton Theory
To characterize the main property of solitons, we will start from KdV equation
(1.1), considering two different limits. The first one is given by the dispersive
linear equation
Ut = Uxxx. (1.2)
The particular solution of this equation is U(x, t) = U0ei(kx−ωt), with the disper-
sion relation ω = k3. Then, the phase velocity of this elementary wave ωk
= k2
depends on k. Due to linearity of the equation any superposition of these waves
U =∑
k akei(kx−k3t) is also a solution of equation(1.2). But since each component
of this ”wave packet” will travel with different velocity (they are dispersive), the
wave will change shape during propagation, and, in general,will spread out.
In another (non-dispersive) limit, nonlinear equation (1.1) has the form
Ut + UUx = 0. (1.3)
To understand the behaviour of a solution of this equation, we will consider first
the simple linear wave equation
Ut + cUx = 0, (1.4)
describing wave, propagating with velocity c, with shape given in terms of an
arbitrary smooth function
U(x, t) = U(x− ct).
Then, a solution of nonlinear equation (1.3) has to appear in unexplicit form,
simply replacing c by function U(x,t), as follows
U(x, t) = U(x− U(x, t)t).
As easy to see, at finite time, the form of Ux becomes infinite and it indicates an
appearance of the shock wave.
But in the KdV equation (1.1) both terms (dispersion and nonlinearity) ap-
pear simultaneously. Therefore, their effects are opposite in character, exactly
compensate one another and provide the stable structure.
In general, solitons are defined to be special solutions of some nonlinear partial
differential equations with the following properties
• Solitons are traveling waves
• The energy of the wave is finite. It is continuous, bounded and localized in
space.
6
• They are stable
• They possesses an elastic collision and keep their identities after pairwise
collisions.
• An initial wave will asymptotically decompose into one or more solitons
depending on amplitude and other properties.
7
Chapter 2
EVOLUTION EQUATIONS AND HAMILTONIAN SYSTEMS
2.1 Hamiltonian Dynamical Systems
The Hamiltonian theory is an important tool in the classical mechanics [79]
and plays crucial role in the soliton theory [11]. Hamilton’s canonical equations
for a system with n degrees of freedom, defined in 2n dimensional phase space,
pi = −∂H(pi, qi)
∂qi, qi =
∂H(pi, qi)
∂pi
, where i = (1, 2, ..., n) (2.1)
represent special case of finite dimensional dynamical systems generated by one
Hamiltonian function H(qi, pi) where q1, ..., qn are generalized coordinates and
p1, ..., pn are generalized momenta. In this phase space the Poisson bracket of
two functions with respect to canonical variables (q, p) is defined as the following
skew-symmetric bilinear form
F,G =n∑
i=1
∂F
∂qi
∂G
∂pi
− ∂F
∂pi
∂G
∂qi, (2.2)
that satisfies the following properties:
1. Skew-Symmetry
F,G = −G,F, F, F = 0 (2.3)
2. Linearity
λF1 + µF2, G = λF1, G+ µF2, G (2.4)
where λ and µ are arbitrary constant
3. Leibnitz Rule
F1F2, G = F1F2, G+ F2F1, G (2.5)
8
4. The Jacobi Identity
F1, F2, F3+ F2, F3, F1+ F3, F1, F2 = 0 (2.6)
The Poisson brackets can be written in a compact form as follows
F,G = (∇F )TJ(∇G) =n∑
i,k=1
∂F
∂Xi
Jik
∂G
∂Xk
, (2.7)
where generalized gradients and symplectic metric are defined as
∇F =
∂F∂X1
.
.
.
∂F∂X2n
, ∇G =
∂G∂X1
.
.
.
∂G∂X2n
, Jn =
0 In
−In 0
2n×2n
(2.8)
where
Xi = (q1, q2, ..., qn, p1, p2, ..., pn). (2.9)
Then, the Hamilton equations(2.1) take the form of 2n dimensional dynamical
system (the gradient system) as
Xi = Jik
∂H
∂Xk
. (2.10)
Definition 2.1.0.1 A function F (qi, pi) is a first integral of Hamiltonian systemwith Hamiltonian function H(qi, pi) if and only if the Poisson bracket H,F = 0.
Definition 2.1.0.2 Two functions F1(qi, pi) and F2(qi, pi) are in involution iftheir Poisson bracket is equal zero, F1, F2 = 0.
Theorem 2.1.0.3 (Liouville) If a system with n degrees of freedom (in 2n di-mensional phase space)admits n independent first integrals of motion in involu-tion then the system is integrable by quadratures.
2.2 Separation of Variables
An integrable system admits separation of variables by canonical transforma-
tion to the action-angle variables. We can represent this by the following diagram
H(qi, pi) −−−−−−−−−−−−−−−−−−−−−→Canonical T ransformation
H(I1, ..., In)
q1, ..., qn ϕ1, ..., ϕn
p1, ..., pn I1, ..., In
Canonical variables Action− angle variables
9
qi = ∂H∂pi
ϕi = ∂H(I)∂Ii
= ωi(I)
pi = −∂H∂qi
Ii = −∂H(I)∂ϕi
= 0
One uses canonical transformation from (q, p) to (ϕ, I) variables, called the
action-angle variables [79], such that Hamiltonian H is a function of only ac-
tion variables Ii. Then, equations of motion in these variables show that action
variables (I1, ..., In) are independent integrals of motion, while angle variables
ϕ1, ..., ϕn are linear functions of time
ϕi = ωit+ ϕi(0).
Using these canonical transformations we can solve Hamiltonian’s dynamics
according to diagram
q(0), p(0) −−−−−−−→Direct CT
I(0), ϕ(0)
↓ ↓q(t), p(t) ←−−−−−−−−
Inverse CTI(t), ϕ(t)
2.3 Evolution Equations and Dynamical Systems
Definition 2.3.0.4 Partial differential equation for function U(x, t) in the form
Ut = F (U,Ux, Uxx, ...)
is called the evolution equation.
Example: The KdV equation
Ut = Uxxx − 6UUx
Definition 2.3.0.5 The first order system of equations in the form
al =
N∑
n=1
Fln(a1, ..., aN)
where al(t), (l = 1, .., N) is a vector field in N-dimensional space, is called thedynamical system.
Example:The Henon-Heiles system
q1 = p1, p1 = −3q21 −
1
2q22
q2 = p2, p2 = −q1q2.
10
2.4 Infinite Dimensional Dynamical Systems
An evolution equation can be considered as a dynamical system in infinite
dimensional space. For example, if function U(x, t) is given in the interval x ∈[0, 2π], then we can expand it to the Fourier series
U(x, t) =
∞∑
n=−∞an(t)einx. (2.11)
Substituting this expansion to the KdV equation (1.1),then multiplying this
equality by e−ilx, taking integral and using definition of the Dirac delta func-
tion (δ(l−m) = 12π
∫ ∞−∞ e−i(l−m)xdx) we obtain the following system of equations
for the Fourier components of U(x, t)
al = (il)3al + 6∞∑
n=−∞(in)anal−n, where (l = 0,±1,±2, ...). (2.12)
It represents an infinite dimensional nonlinear dynamical system. Moreover, this
dynamical system is the Hamiltonian dynamical system in the form of Eq. (2.10).
2.5 KdV as a Hamiltonian System
The KdV equation can be written as the Hamiltonian system
Ut = −∂(3U2 − Uxx)
∂x=
∂
∂x
δH
δU(x), (2.13)
where the Hamiltonian functional H is
H [U ] = −∫ ∞
−∞(U2
x
2+ U3)dx, (2.14)
and symbol δ/δU denotes variational derivative. The Poisson bracket in this
case, called the Faddeev-Zakharov [7] bracket, is defined as follows
S,R =
∫ ∞
−∞
δS
δU(x)
∂
∂x
δR
δU(x)dx, (2.15)
where ∂/∂x is the skew symmetric operator.
If ∂∂x
δHδU(x)
is expressed as
∫ ∞
−∞δ(x− y) δH
δU(y)dy = U,H, (2.16)
then equation (2.13) becomes in Hamiltonian form,
∂U
∂t= U,H. (2.17)
11
Thus the KdV can be considered as the Hamiltonian dynamical system. Moreover
it admits the second Hamiltonian structure
Ut = (∂3x − 2U∂x − 2∂xU)
δ
δU(x)
∫ ∞
−∞
1
2U2dx, (2.18)
with Magri bracket [130] and this shows bi-Hamiltonian nature of KdV equation.
Furthermore, it is integrable Hamiltonian system. But to discuss infinite
dimensional dynamical integrable systems we need an infinite number of integrals
of motion (I1, ..., In) in involution. Then, according to Liouville theorem the
system is formally integrable.
For the KdV equation following functionals
I1 =
∫ ∞
−∞U(x)dx (2.19)
I2 =
∫ ∞
−∞U2(x)dx (2.20)
I3 =
∫ ∞
−∞(U2
x
2+ U3)dx (2.21)
I4 =
∫ ∞
−∞(U2
xx
4− UUx + U4)dx (2.22)
...
are integrals of motion, which are in involution according to the Faddeev-Zakharov
bracket (2.15), In, Im = 0. So the system is integrable.
2.6 Spectral Transform for Linear equations
For separation of variables in this system most powerful method is the Inverse
Scattering Method(ISM) [5] or the Nonlinear Fourier transform [19].
To illustrate the idea let us consider Fourier transform for the linear equation
(1.2)(Initial Value Problem)
Ut = Uxxx, U(x, 0) = F (x); (2.23)
U(x, t) =1
2π
∫ ∞
−∞eikxU(k, t)dk FourierTransform (2.24)
U(k, t) =
∫ ∞
−∞e−ikxU(x, t)dx InverseFourierTransform (2.25)
Then,the Initial Value Problem is solved by the following diagram
U(x, 0) −−−−−−−−−−−−−−−→Fourier Transform
U(k, 0)
↓ ↓ Linear time evolution
U(x, t) ←−−−−−−−−−−−−−−−−−−−−−−Inverse Fourier Transform
U(k, t)
12
2.7 Nonlinear Fourier Transform
Similarly works the nonlinear Fourier transform for KdV equation [15]
U(x, 0) −−−−−−−−−−−−−−−−−−−−−→Direct Spectral T ransform
S(0)
↓ ↓U(x, t) ←−−−−−−−−−−−−−−−−−−−−−−
Inverse Spectral T ransformS(t)
where S(t) is Nonlinear Fourier image.
2.8 Linearization of Nonlinear Problem
To develop the spectral transform to a Nonlinear evolution equation one needs
the so called linear representation. This auxiliary linear problem is known as the
Lax pair representation [8]. For the KdV equation it can be represented by the
following linear system
−φxx = (λ− U)φ, (2.26)
−φt = −4∂3φ
∂x3+ 3Uφx + Uxφ+ φxU, (2.27)
where φ = φ(x, λ, t) and λ is a spectral parameter.
Let L be the Schrodinger operator
L = − ∂2
∂x2+ U, (2.28)
and A be
A = −4∂3
∂x3+ 3(U
∂
∂x+
∂
∂xU). (2.29)
If we write equations (2.26) and (2.27) in terms of these operators we get the
system
Lφ = λ2φ, (2.30)
φt = Aφ, (2.31)
which is called the Lax representation. If isospectrality condition satisfies (which
means that ∂λ/∂t = 0), then KdV equation is equivalent to the operator equation
Lt = [A,L]. (2.32)
In general many nonlinear partial differential equations which are integrable are
related to existence of the Lax pair [16].
Another form of linear representation relates to commutativity (compatibil-
ity) condition
Ut − Vx + [U, V ] = 0, (2.33)
13
for the following linear system of equations
∂ψ
∂x= Uψ, (2.34)
∂ψ
∂t= V ψ. (2.35)
This form of the linear problem is known as the zero-curvature representation
[5], [20]. And integrable hierarchy can be naturally developed in this approach
[19].
14
Chapter 3
AKNS INTEGRABLE HIERARCHY
3.1 Hierarchy of Integrable Evolutions
The hierarchy of integrals of motion ( 2.19- 2.22) for the KdV equation, gen-
erates hierarchy of nonlinear evolution equations in the form
Utn = U, In, (3.1)
with Faddeev-Zakharov bracket (2.15). In can be considered as Hamiltonians
of corresponding evolution equations in times t1, ..., tn, ....The hierarchy of these
equations is related to the hierarchy of corresponding linear problems.
3.2 AKNS Hierarchy
In 1974 American mathematicians (Ablowitz, Kaup, Newell, Segur) introduced
the AKNS Hierarchy of nonlinear evolution equations [19]. This hierarchy in-
cludes several nonlinear evolution equations as the Nonlinear Schrodinger equa-
tion, and Modified KdV equation, as special cases of equations of degree two and
three respectively.
The AKNS hierarchy arises from a sequence of linear evolutions
∂ψ
∂tn= Vnψ, (n = 0, 1, 2, ...) (3.2)
and the Zakharov-Shabat spectral problem
∂ψ
∂x=
λ q
r −λ
ψ = Uψ. (3.3)
The ψ is a vector (ψ1, ψ2)T and q, r are potentials. Differentiating these equations
∂
∂x(∂ψ
∂tn) = (Vn)xψ + Vnψx = (Vn)xψ + VnUψ, (3.4)
15
∂
∂tn(∂ψ
∂x) = Utnψ + Uψtn = Utnψ + UVnψ, (3.5)
we get compatibility condition for the system (3.2),(3.3) in the form
∂U
∂tn− ∂Vn
∂x+ UVn − VnU = 0. (3.6)
Similarly, by the compatibility of tn and tm evolutions (3.2)
∂tn∂tmψ = ∂tm∂tnψ (3.7)
we have
∂Vm
∂tn− ∂Vn
∂tm+ VnVm − VmVn = 0. (3.8)
To construct AKNS hierarchy let us suppose that U,V in equations(2.34),(2.35)
have the form
V =
a(λ) b(λ)
c(λ) −a(λ),
, U =
λ q
r −λ
, (3.9)
Then substituting in equation (2.33)
Ut − Vx + [U, V ] = 0, (3.10)
we can find the following restrictions on functions a, b, c,
ax = qc− rb,bx − 2λb = qt − 2qa,
cx + 2λc = rt + 2ra.
. (3.11)
We assume VN as a polynomial in spectral parameter λ degree N . It implies
the following equations
a =N∑
n=0
λnan, b =N∑
n=0
λnbn, c =N∑
n=0
λncn. (3.12)
Substituting a, b, c to the equation (3.11) we have the recurrence relations as
b0x = qt − 2 q a0,
c0x = rt − 2 r a0,
a0x = q c0 − r b0;
(3.13)
16
bnx − 2 bn−1 = −2 q an,
cnx + 2 cn−1 = 2 r an,
anx = q cn − r bn. (n = 1, 2, ..., N)
(3.14)
Solving the last equation as an =∫ x
(q cn−r bn) and substituting it to first couple
of equations (3.13) we get
qt = b0x+ 2 q
∫ x(q c0 − r b0),
rt = c0x− 2 r
∫ x(q c0 − r b0).
(3.15)
These equations can be written in the matrix form as
q
r
t
= R
b0
c0
, (3.16)
where integro-differential matrix operator R is
R =
∂x − 2q
∫ xr 2q
∫ xq
2r∫ x
r ∂x − 2r∫ xq
. (3.17)
The first couple of equations (3.14) in terms of R is
bn−1
cn−1
=1
2σ3R
bn
cn
, n = 1, ..., N , (3.18)
and recursively we have
b0
c0
= ℜN
bN
cN
, (3.19)
where ℜ = 12σ3R is called the recursion operator. This recursion operator
generates the hierarchy of evolutions
1
2σ3
q
r
tN
= ℜ
b0
c0
= ℜN+1
bN
cN
. (3.20)
To have equations in a closed form let us fix bN , cN in the simplest form
bN
cN
=
q
r
.
17
Then we find the hierarchy of evolution equations
1
2σ3
q
r
tN
= ℜN+1
q
r
. (3.21)
For particular values of N we have equations;
for N=0
qt0 = qx,
rt0 = rx;(3.22)
for N=1
qt1 = 12qxx − q2 r,
rt1 = −12rxx + r2 q;
(3.23)
for N=2
qt2 = 14qxxx − 1
2(q2r)x + 1
2q2rx − 1
2qrqx,
rt2 = 14rxxx − 1
2(r2q)x + 1
2r2qx − 1
2rqrx.
(3.24)
By the following identification
q =√
−λ8e+,
r =√
−λ8e−,
(3.25)
the recursion operator ℜ (integro differential ) becomes
ℜ =
∂x − λ
4e+
∫ xe− −λ
4e+
∫ xe+
−λ4e−
∫ xe− ∂x + λ
4e−
∫ xe+
. (3.26)
Then the first three members of AKNS hierarchy (3.22), (3.23), (3.24) in terms
of e+ and e− appear as
∂t0e± = ∂xe
±; (3.27)
±∂t1e± = ∂2
xe± +
λ
4e+e−e±; (3.28)
∂t2e± = ∂3
xe± +
3λ
4e+e−∂xe
±. (3.29)
The first couple of equations (3.27) are the linear wave equations. The second
system (3.28) is called the Reaction-Diffusion system [20]. It is connected with
low dimensional gravity, constant curvature surfaces and quantum theory and
has been studied in [76]. The last system with cubic dispersion has reductions
to KdV, MKdV and mixed KdV-MKdV equations.
18
3.3 KdV and MKdV reductions
The third member of hierarchy (3.29) admits following reductions [20].
1) The first reduction e+ = U, e− = 1, leads to the KdV equation
∂t2U = ∂3xU +
3λ
4U∂xU. (3.30)
Under this reduction the reduced hierarchy (3.21) becomes
∂
∂t2k
U
0
= ℜkKdV ∂x
U
0
, (3.31)
or in the scalar form, the KdV hierarchy
∂t2kU = Rk
KdV (∂xU), (3.32)
with the recursion operator of the KdV hierarchy given by [130]
RKdV = (∂21 +
λ
2U +
λ
4∂1U
∫ x
). (3.33)
2) Under the second reduction, e+ = e− = U , we obtain MKdV equation
∂t2U = ∂3xU +
3λ
4U2∂xU, (3.34)
and the corresponding reduced hierarchy
∂
∂t2k
U
U
= ℜkMKdV ∂x
U
U
. (3.35)
In the scalar form it gives MKdV hierarchy
∂t2kU = Rk
MKdV (∂xU), (3.36)
with the recursion operator
RMKdV = (∂21 +
λ
2U2 +
λ
2∂1U
∫ x
U). (3.37)
3.4 The mixed KdV-MKdV hierarchy
We will consider here also a new reduction of system (3.29) in the form
e+ = (α + β)U
e− = αU + β(3.38)
19
where α, β are arbitrary real constants.
For this mixed case, the reduced equation is of the form of mixed KdV-MKdV
equation
∂t2U = ∂3xU +
3λ
4(α + β)(αU2∂xU + β U∂xU). (3.39)
Then, the corresponding hierarchy (3.21)can be reduced to
∂
∂t2k
(α + β)U
−αU
= ℜ2kmix∂x
(α + β)U
−αU
, (3.40)
or
(α + β)
−α
∂
∂t2k
U =
(α+ β)
−α
Rkmix∂xU (3.41)
and in scalar form
∂t2kU = Rk
mix∂xU (3.42)
The recursion operator corresponding to this mixed KdV and MKdV hierar-
chy is
Rmix = (∂21 +
λ
2(α + β)U(αU + β) +
λ
4(α + β)∂1U
∫ x
2αU + β). (3.43)
In particular cases it reduces to recursion operators
a)α = 0, β = 1 ⇒ MKdV
b)α = 1, β = 0 ⇒ KdV(3.44)
20
Chapter 4
GENERALIZED HENON-HEILES SYSTEM
4.1 Henon-Heiles system
One of the most popular model to study integrability and chaos is called the
Henon-Heiles system. Introduced by Henon and Heiles in 1964 [83], this system
describes the motion of a star in the gravitational field of a galaxy. The model
describes two one-dimensional harmonic oscillators with a cubic interaction and
has been discussed in applications to the cosmic rays [101] , for the oscillations
of atoms in a three-atomic molecule [86], for geodesic flows on SO(4) [84] and
three-particle Toda lattice theory [85].
The generalized form of the Henon-Heiles system is
q1 + c1q1 = bq21 − aq2
2 ,
q2 + c2q2 = −2aq1q2,(4.1)
and its energy is given as
E = (1
2q1
2 + q22 + c1q
21 + c2q
22) + aq1q
22 −
1
3bq31. (4.2)
This model is an example of Hamiltonian system with a mixed phase space
structure, i.e., partially ordered and partially chaotic. For generic parameters
a, b, c, the system possesses chaotic orbits and the energy (4.2) is the only con-
served quantity. By increasing the total energy, a transition from an integrable
to an ergodic system is induced. This model, firstly introduced to describe the
chaotic motion of stars in a galaxy, it later became an important milestone in
the development of the theory of chaos [100], partly because of the conceptual
simplicity of the model.
On the other hand, the system was found to have a second independent
integral of motion and to be integrable only for some fixed values of parameters
[94]. The integrable cases of the Henon-Heiles system (4.1)are
21
1.
a/b = −1 , c1 = c2; (4.3)
2.
a/b = −1/6 , c1, c2 arbitrary; (4.4)
3.
a/b = −1/16 , c1 = 16c2. (4.5)
At the end of 1970’s, very important discovery was made. Bogoyavlenskii and
Novikov [91] showed that each of the stationary reductions of the KdV hierar-
chy constitutes a completely integrable, finite dimensional Hamiltonian system.
Then Fordy [98] has observed that the above integrable cases of the Henon-Heiles
system are closely related to stationary flows [90], [99] of integrable fifth-order
nonlinear evolution equations, including the higher KdV equation.
4.2 Stationary reductions from Soliton Equations
The above integrable cases of Henon-Heiles system can be reduced from soliton
equations as 5-th KdV, Sawada-Kotera and Kaup-Kupershmidt equations. To
illustrate this relation we consider equations (4.1), where for simplicity we take
c1 = c2 = 0. Differentiating twice the first equation of the Henon-Heiles system
(4.1) and eliminating q2 variable by the use of the second equation of the system,
and the energy equation (4.2), we get for q1 the fourth order equation
¨q1 = 2(a+ b)q21 − 4aE +
20
3abq3
1 + (2a− 8b)q1q1. (4.6)
This equation by following identification
U = q1, Ux = q1, Uxx = q1, ... (4.7)
has form of the fourth-order ordinary differential equation for U(x):
Uxxxx − 2(a+ b)U2x −
20
3abU3 + (8a− 2b)UUxx = −4aE. (4.8)
The last one is the stationary reduction of the general evolution equation
Ut = (Uxxxx − 2(a+ b)U2x −
20
3abU3 + (8a− 2b)UUxx)x. (4.9)
The known integrable reductions of the last equation are:
1. the Sawada-Kotera equation (a = 1/2, b = −1/2)
Ut = (Uxxxx + 5UUxx +5
3U3)x (4.10)
22
2. the fifth-order KdV equation (a = 1/2, b = −3)
Ut = (Uxxxx + 10UUxx + 5U2x + 10U3)x (4.11)
3. the Kaup-Kupershmidt equation (a = 1/4, b = −4)
Ut = (Uxxxx + 10UUxx +15
2U2
x +20
3U3)x (4.12)
The stationary reductions of these three equations coincide with corresponding
integrable cases (4.3)-(4.5) of the Henon-Heiles model (5.1).
4.3 The Hamilton-Jacobi theory
In the Hamilton-Jacobi theory canonical transformations may be used to pro-
vide a general procedure for solving mechanical problems [79]. If we consider
a canonical transformation from coordinate and momenta (q(t),p(t)) at time t
to new set of constant quantities (Q(t),P(t)) which may be 2n initial values
Q = q0, P = p0 at time t=0, then equations of transformation relating the old
and the new canonical variables give desired solution of the mechanical problem:
q = q(q0, p0, t) (4.13)
p = p(q0, p0, t) (4.14)
Since new variables are constant in time, it requires the transformed Hamiltonian
H to be identically zero
∂H
∂Pi
= Qi = 0 (4.15)
∂H
∂Qi
= Pi = 0 (4.16)
If the generating function S(q, P, t) is a function of old coordinates q and new
momenta P , canonical transformations are given by formulas
pi =∂S
∂qi, Qi = − ∂S
∂Pi
, (4.17)
H = H +∂S
∂t. (4.18)
Since H = 0, from the last equation, after substituting old momenta according
to the first Eq. (4.17), we have the Hamilton-Jacobi equation
23
H(q1, ..., qn;∂S
∂q1, ...,
∂S
∂qn; t) +
∂S
∂t= 0 (4.19)
This equation is the first order partial differential equation (PDE) in (n + 1)
variables q1, ..., qn, t. Solution of this equation, function S, is Hamilton’s Principal
function. Complete solution of Eq.(4.19) as the first order PDE, depends on
(n+ 1) constants of integration α1, ..., αn+1:
S = S(q1, ..., qn, α1, ..., αn+1, t). (4.20)
But since only derivatives of S (but not S itself )are involved in the Hamilton-
Jacobi equation, one constant is irrelevant. Indeed, if S is some solution of the
equation, then S+α is also solution, where α is an arbitrary constant. It appears
as an additive constant. Therefore complete solution with non-additive constants
can be written in the form
S = S(q1, ..., qn, α1, ..., αn, t). (4.21)
We can take n constants of integration to be new (constant)momenta
Pi = αi, i = 1, ..., n . (4.22)
At time t0
pi =∂S(q, α, t)
∂qi,
these constitute n-equations relating n α’s with initial q and p. Other half of
equations of transformation
Qi = βi =∂S(q, α, t)
∂αi
,
provides new constant coordinates βi at time t0 which can be obtained from
initial conditions with known initial values of qi. Then, expressions
qj = qj(α, β, t), (4.23)
pi = pi(α, β, t), (4.24)
solves the problem giving coordinates and momenta as functions of time and
initial conditions. Hamilton’s Principal function as the generator of a canonical
transformation to constant coordinates and momenta has physical meaning of
the action.
24
4.4 Separation of Variables
Separation of variables is an efficient method of solving Hamilton-Jacobi equa-
tions [79].
Separation of Time Variable
If Hamiltonian H is not an explicit function of t
∂H
∂t= 0, (4.25)
the time variable can be separated in the Hamilton-Jacobi equation. Assuming
solution in the form
S(q, t) = W (q)− α1t. (4.26)
we have reduced equation
H(q,∂W
∂q) = α1. (4.27)
From this equation, one constant of integration in S, namely α1, is thus equal to
the constant value of H. Here time independent function W is Hamilton’s charac-
teristic function. This function generates canonical transformation in which all
new coordinates are cyclic
Pi = − ∂H∂Qi
= 0, Pi = αi (4.28)
Because the new Hamiltonian depends on only one of the momenta P1 = α1, the
equations of motion are
Qi =∂H
∂αi
=
1, i = 1
0, i 6= 1(4.29)
Then we have solution
Q1 = t+ β1 =∂Wi
∂αi
, (4.30)
Qi = βi =∂Wi
∂αi
i 6= 1. (4.31)
It shows that the only coordinate that is not simply a constant of motion is Q1.
Generalizing, instead of α1 and constants of integration as a new momenta,
one can choose new momenta as an independent functions γ1, ..., γn:Pi = γi(αi, ..., αn).
Then, characteristic function W can be expressed in terms of qi and γi as inde-
pendent variables:
W = W (qi, γi)
25
Thus, equations of motion become
Qi =∂H
∂γi
= νi(γ), (4.32)
and in this case all new coordinates are linear functions of time
Qi = νit+ βi. (4.33)
Separable Hamilton-Jacobi Equation
If the Hamilton-Jacobi equation admit a separating variable then solution can
be reduced to quadratures.
Definition 4.4.0.6 Coordinate q1 is said to be separable in Hamilton-Jacobiequation, when Hamilton’s principal function can be split into two additive parts
S(q1, ..., qn, α1, ..., αn, t) = S1(q1, α1, ..., αn, t) + S ′(q2, ..., qn, α1, ..., αn, t). (4.34)
In this case the Hamilton-Jacobi equation can be split in two equations, the first
for S1 and the second for S ′.
Definition 4.4.0.7 Hamilton-Jacobi equation is completely separable if all co-ordinates in problem are separable. Hamilton’s Characteristic function for com-pletely separable problem has the form
W (q1, ..., qn, α1, ..., αn) =n∑
i=1
Wi(qi, α1, ..., αn) (4.35)
For this solution the Hamilton-Jacobi equation will split into n equations
Hi(qi,∂W
∂qi, α1, ..., αn) = αi , (i = 1, ..., n) (4.36)
where αi are separation constants. This set of first order ordinary differential
equations is always reducible to quadratures. We solve it first for ∂S/∂qi and
then integrate over qi.
26
Chapter 5
SEPARATION OF VARIABLES IN H-H SYSTEM
The Henon-Heiles model (4.1) in the integrable case 2, Eq. (4.4), where a =
1/2, b = −3, c1 = c2 = 0, is two dimensional Hamiltonian dynamical system
q1 = −3q21 − 1
2q22,
q2 = −q1q2.(5.1)
The Hamiltonian function H
H =p2
1
2+p2
1
2+q1q
22
2+ q3
1 (5.2)
is the first integral of motion of the system. The system admits in addition the
second integral of motion F as
F = −2q2p1p2 + 2q1p22 −
1
4q42 − q2
1q22. (5.3)
According to Liouville theorem for integrability of this system one needs two
independent integrals of motion H and F . Then choosing these integrals as a
new momenta and performing corresponding canonical transformation one can
represent the system in the action-angle variables. But explicit realization of this
program can be done only in some particular cases. The first step in this real-
ization is separation of variables in the Hamilton-Jacobi equation [77], [80], [89],
[92]. For this separation we need to find proper canonical transformation from
original coordinates q1, q2 to new coordinates Q1, Q2 with generating function
F(p,Q), where p1, p2 are old momenta. Then using
Pi =∂F∂Qi
, qi =∂F∂pi
, (i = 1, 2)
one can derive the old momenta p1, p2 in terms of the new coordinates Q1, Q2 and
conjugate momenta P1, P2. Rewriting conserved functions h1 = H and h2 = F in
27
terms of new coordinates and new momenta, we have to solve the corresponding
Hamilton-Jacobi equations
hi = hi(Qi,∂Wi
∂Qi
, t) = αi , (i = 1, 2)
for Hamilton’s characteristic functions Wi (i = 1, 2). Our problem is separable
in new coordinates if one can represent Wi as
W1 = W11(Q1, C1, C2) +W12(Q2, C1, C2)
W2 = W21(Q1, C1, C2) +W22(Q2, C1, C2)(5.4)
where C1, C2 are constants, and reduce problem to findWik as solution of ordinary
differential equation.
5.1 Bi-Hamiltonian Formulation
The Hamilton equations (2.1) in the matrix form(2.10) for the Henon-Heiles
system (5.1) are
~x =
p1
p2
q1
q2
t
= J0∇H, (5.5)
where J0 is a constant symplectic matrix
J0 =
0 0 1 0
0 0 0 1
−1 0 0 0
0 −1 0 0
. (5.6)
For the Henon-Heiles model we have the second integral of motion F given by
Eq.(5.3). The similar form satisfies for F
~x =
p1
p2
q1
q2
t
= J1∇F (5.7)
28
where J1 is a skew symmetric matrix
J1 =1
q22
0 0 q112q2
0 0 12q2 0
−q1 −12q2 0 1
2p2
−12q2 0 −1
2p2 0
, (5.8)
so that F is also Hamiltonian and this can be rewritten as
J1∇F = J0∇H = ~x. (5.9)
The system having two Hamiltonians and satisfying this equality is called bi-
Hamiltonian [80]. This shows that the Henon-Heiles model is bihamiltonian
dynamical system [72], [95]. Since F is integral of motion, according to the first
Hamiltonian structure it is in involution with Hamiltonian H. Then the Henon-
Heiles system is integrable by the Liouville theorem. Bi-Hamiltonian structure
means in addition that evolutions generated by two integrals of motions as inde-
pendent momenta, with corresponding Poisson structures, are the same.
5.2 Integrable Extensions
Now we consider several new integrable extensions of the generalized Henon-
Heiles system 5.1 and corresponding separation of variables.
5.2.1 C-Extended Henon-Heiles System
For the case 2 (a/b = −1/6) in equation 5.1 let c1 = c2 = 0, b = 3, a = −1/2.
Then the first extended version with an additional constant term C is
q1 = −3q21 − 1
2q22 + C,
q2 = −q1q2.(5.10)
The corresponding Hamiltonian is
H =p2
1
2+p2
1
2+q1q
22
2+ q3
1 − Cq1 (5.11)
while additional integral of motion F is given by
F = −2q2p1p2 + 2q1p22 −
1
4q42 − q2
1q22 + Cq2
2. (5.12)
Hamiltonian form of these equations with Hamiltonian function H is defined
in extended five dimensional phase space, where C is considered as an extra
29
dynamical variable, and are given in the form
q1
q2
p1
p2
C
t
=
p1
p2
−3q21 − 1
2q22 + C
−q1q20
=
=
0 0 1 0 0
0 0 0 1 0
−1 0 0 0 0
0 −1 0 0 0
0 0 0 0 0
︸ ︷︷ ︸
J0
3q21 + 1
2q22 − C
q1q2
p1
p2
−q1
︸ ︷︷ ︸
∇H
, (5.13)
while the form with F as a Hamiltonian
q1
q2
p1
p2
C
t
=1
q22
0 0 q112q2 0
0 0 12q2 0 0
−q1 −12q2 0 1
2p2 0
−12q2 0 −1
2p2 0 0
0 0 0 0 0
︸ ︷︷ ︸
J1
×
−12p2
2 + 12q1q
22
12p1p2 + 1
4q32 + 1
2q21q2 − 1
2q2C
12q2p2
12q2p1 − q1p2
−14q22
︸ ︷︷ ︸
∇F
. (5.14)
We notice that due to the odd dimension of our extended phase space the skew-
symmetric matrices J0 and J1 are degenerate. Both systems (5.2.1), (5.14) can
be rewritten as
J0∇H = J1∇F = ~x. (5.15)
which means the system (5.10) is bi-Hamiltonian system.
To separate variables in our system we consider Hamilton’s characteristic
function F(p1, p2, Q1, Q2) in the form
30
F = p1(Q1 +Q2) + 2p2
√
−Q1Q2. (5.16)
The canonical transformations generated by this function can be written as
q1 = Q1 +Q2, p1 = P1Q1−P2Q2
Q2−Q1, (5.17)
q2 = 2√−Q1Q2 , p2 =
√−Q1Q2(P1−P2)
(Q2−Q1). (5.18)
Then Hamiltonian H and the second integral of motion F in terms of new
coordinates and momenta are
H =1
2(Q2 −Q1)(P 2
2Q2 − P 21Q1) + (Q1 +Q2)(Q
21 +Q2
2)− C(Q1 +Q2), (5.19)
F = − 2
(Q2 −Q1)(P 2
1 − P 22 )Q1Q2 + 4Q1Q2(Q
21 +Q1Q2 +Q2
2 − C). (5.20)
The Hamilton-Jacobi equation for the first Hamiltonian H is
1
2(Q2 −Q1)((∂W1
∂Q2
)2Q2 − (∂W1
∂Q1
)2Q1) + (Q1 +Q2)(Q21 +Q2
2 − C) = α1 (5.21)
where W function is denoted as W1. For the second Hamiltonian F the Hamilton-
Jacobi equation is
− 2
(Q2 −Q1)((∂W2
∂Q1)2−(
∂W2
∂Q2)2)Q1Q2+4Q1Q2(Q
21+Q1Q2+Q
22−C) = α2 (5.22)
In this case W function is denoted as W2. We try to separate solution of these
equations simultaneously by substitution
W1 = W11(Q1, C1, C2) +W12(Q2, C1, C2),
W2 = W21(Q1, C1, C2) +W22(Q2, C1, C2).(5.23)
Then, it is obvious that new momenta are given by
Pi =∂Wi
∂Qi
, (i = 1, 2). (5.24)
So we have separated solution in quadratures
∂W1k
∂Qk
=
√1
Qk
(2K1 + 2CQ2k − 2Q4
k + 2α1Qk), (k = 1, 2), (5.25)
∂W2k
∂Qk
=
√1
Qk
(2K2Qk + 2CQ2k − 2Q4
k −α2
2), (k = 1, 2). (5.26)
31
5.2.2 D-Extended Henon-Heiles System
The second extension of model(5.1),(4.4)(c1 = 0, c2 = 0)is
q1 = −3q21 − 1
2q22
q2 = −q1q2 + Dq32
(5.27)
where D is an arbitrary constant. The corresponding Hamiltonian H and integral
F are
H =p2
1
2+p2
1
2+q1q
22
2+ q3
1 +D
2q22
(5.28)
F = −2q2p1p2 + 2q1p22 −
1
4q42 − q2
1q22 +
2Dq1q22
. (5.29)
Bi-Hamiltonian representation of system (5.27) in extended phase space is
q1
q2
p1
p2
D
t
=
p1
p2
−3q21 − 1
2q22
−q1q2 + Dq32
0
=
=
0 0 1 0 0
0 0 0 1 0
−1 0 0 0 0
0 −1 0 0 0
0 0 0 0 0
︸ ︷︷ ︸
J0
3q21 + 1
2q22
q1q2 − Dq32
p1
p2
12q2
2
︸ ︷︷ ︸
∇H
, (5.30)
and
q1
q2
p1
p2
D
t
= − 1
2q22
0 0 0 e −f0 0 e −g −h0 −e 0 i −k−e g −i 0 l
f h k −l 0
︸ ︷︷ ︸
J1
32
×
−12p2
2 + 12q1q
22
12p1p2 + 1
4q32 + 1
2q21q2 − 1
2q2C
12q2p2
12q2p1 − q1p2
−14q22
︸ ︷︷ ︸
∇F
(5.31)
where
e =1
q2, f = 2q2p2, g =
2q1q22
, h = (2q2p1 − 4q1p2), (5.32)
i =p2
q22
, k = (−2q1q22 +2p2
2+2D
q22
), l = 2p1p2+q32 +2q2
1q2+4Dq1q32
(5.33)
and
J0∇H = J1∇F = ~x. (5.34)
Now we will do the canonical transformation that is given in the form (5.16),
(5.17), (5.18) as for the first extension. After transformation, the Hamiltonians
H and F become as
H =1
2(Q2 −Q1)(P 2
2Q2 − P 21Q1) + (Q1 +Q2)(Q
21 +Q2
2)−D
4Q1Q2
, (5.35)
F = − 2
(Q2 −Q1)((P 2
1−P 22 )Q1Q2)+4Q1Q2(Q
21+Q1Q2+Q
22)−
D
2Q1− D
2Q2. (5.36)
The Hamilton-Jacobi equation defined by these two Hamiltonians are separable
in the form (5.23). We find the separated equations as
∂W1k
∂Qk
=
√
1
Qk
(2K1 −D
2Qk
+ 2Q4k + 2α1Qk), (k = 1, 2), (5.37)
∂W2k
∂Qk
=
√
1
Qk
(2K2Qk − 2Q4k −
α2
4− D
4Qk
), (k = 1, 2). (5.38)
5.2.3 The Harmonic Extension
The third extension of the system (5.1),(4.4) with arbitrary c1 6= 0 and c2 6= 0
constants, which we denote as c1 = a and c2 = b is
q1 + aq1 = −3q21 − 1
2q22,
q2 + bq2 = −q1q2.(5.39)
33
The corresponding Hamiltonians
H =p2
1
2+p2
1
2+aq2
1
2+bq2
2
2+q1q
22
2+ q3
1 ; (5.40)
and
F = −2q2p1p2 + 2q1p22 −
1
4q42 − q2
1q22 − 2bq2
2q1 − (4b− a)(p22 + bq2
2), (5.41)
determine bi-Hamiltonian systems respectively
q1
q2
p1
p2
t
=
p1
p2
−3q21 − 1
2q22 − aq1
−q1q2 − bq20
=
=
0 0 1 0
0 0 0 1
−1 0 0 0
0 −1 0 0
︸ ︷︷ ︸
J0
3q21 + 1
2q22 + aq1
q1q2bq2
p1
p2
︸ ︷︷ ︸
∇H
, (5.42)
q1
q2
p1
p2
t
=1
2q22
0 0 k l
0 0 l 0
−k −l 0 m
−l 0 −m 0
︸ ︷︷ ︸
J1
×
2p22 − 2q1q
22 − 2bq2
2
−2p1p2 − q32 − 2q2
1q2 − 4bq2q1 − 2b(4b− a)q2−2q2p2
−2q2p1 + 4q1p2 − 2(4b− a)p2
︸ ︷︷ ︸
∇F
, (5.43)
where
k = −2q1 − (4b− a), l = −q2, m = −p2. (5.44)
Canonical transformation and the characteristic function in this case includes
additional terms depending on a and b.
34
F = p1(Q1 +Q2 + 4b− a) + 2p2
√
−Q1Q2, , (5.45)
q1 = Q1 +Q2 + (4b− a), p1 = P1Q1−P2Q2
Q2−Q1
, (5.46)
q2 = 2√−Q1Q2, p2 =
√−Q1Q2(P1−P2)
(Q2−Q1). (5.47)
Hamiltonians in terms of new variables are
H =1
2(Q2 −Q1)(P 2
1Q1 − P 22Q2) + [(4b− a)(12b− a)](Q1 +Q2)
4(5.48)
+(6b− a)(Q21 +Q2
2 +Q1Q2) + (Q1 +Q2)(Q21 +Q2
2) +a
2(4b− a
2)2,
F =2
(Q2 −Q1)(P 2
1 − P 22 )Q1Q2 + 4Q1Q2(Q
21 +Q1Q2 +Q2
2) (5.49)
(4b− a)(12b− a)Q1Q2 + 4(6b− a)(Q1 +Q2)Q1Q2.
Then, the Hamilton-Jacobi equations determined by these Hamiltonians are sep-
arated in the next form
∂W1k
∂Qk
=
√1
2Qk
(
4K1 − (4b− a)(12b− a)Q2k − 4(6b− a)Q3
k − 4Q4k (5.50)
+2(2α1 − b(4b− 2a)2)Qk
) 1
2
,
∂W2k
∂Qk
=
√
4
2Qk
[K2Qk −Q4k − (6b− a)Q3
k −(4b− a)
4(12b− a)Q2
k −α2
2]. (5.51)
where (k = 1, 2)
5.2.4 The Harmonic C-Extension
The fourth generalization is an arbitrary mixture of the first and the third
extensions (5.10),(5.39) with arbitrary constants a, b, C
q1 + aq1 = −3q21 − 1
2q22 + C,
q2 + bq2 = −q1q2.(5.52)
35
Hamiltoninans for this case are
H =p2
1
2+p2
1
2+q1q
22
2+ q3
1 +aq2
1
2+bq2
2
2− Cq1, (5.53)
F = −2q2p1p2 + 2q1p22 −
1
4q42 − q2
1q22 − 2bq2
2q1 − (4b− a)(p22 + bq2
2) + Cq22. (5.54)
Bi-Hamiltonian form determined by the first Hamiltonian is given by the
following system
q1
q2
p1
p2
C
t
=
p1
p2
−3q21 − 1
2q22 − aq1 + C
−q1q2 − bq20
=
=
0 0 1 0 0
0 0 0 1 0
−1 0 0 0 0
0 −1 0 0 0
0 0 0 0 0
︸ ︷︷ ︸
J0
3q21 + 1
2q22 + aq1 − C
q1q2 + bq2
p1
p2
−q1
︸ ︷︷ ︸
∇H
, (5.55)
while the form determined by the second Hamiltonian is
q1
q2
p1
p2
C
t
=1
2q22
0 0 k l 0
0 0 l 0 0
−k −l 0 m 0
−l 0 −m 0 0
0 0 0 0 0
︸ ︷︷ ︸
J1
36
×
2p22 − 2q1q
22 − 2bq2
2
−2p1p2 − q32 − 2q2
1q2 − 4bq2q1 + 2[C − b(4b− a)]q2−2q2p2
−2q2p1 + 4q1p2 − 2(4b− a)p2
q22
︸ ︷︷ ︸
∇F
, (5.56)
where
k = −2q1 − (4b− a), l = −q2, m = −p2. (5.57)
Canonical transformation and characteristic function of the form (5.45),(5.46),(5.47)
gives for Hamiltonians the next expressions
H =1
2(Q1 −Q2)(P 2
1Q1 − P 22Q2) + [(4b− a)(12b− a)− 4C]
(Q1 +Q2)
4(5.58)
+(6b− a)(Q21 +Q2
2 +Q1Q2) + (Q1 +Q2)(Q21 +Q2
2) + (4b− a
2)[(4b− a)b− C],
and
F =2
(Q2 −Q1)(P 2
1 − P 22 )Q1Q2 + 4Q1Q2(Q
21 +Q1Q2 +Q2
2) (5.59)
[(4b− a)(12b−−a)− 4C]Q1Q2 + 4(6b− a)(Q1 +Q2)Q1Q2.
Then, separation of Hamilton-Jacobi equations appear as following ODEs
∂W1k
∂Qk
=
√1
2Qk
(
4K1 − [(4b− a)(12b− a)− 4C]Q2k − 4(6b− a)Q3
k (5.60)
−4Q4k + (4α1 − E)Qk
) 1
2
,
∂W2k
∂Qk
=
√1
2Qk
(
4K2Qk − 4Q4k − 4(6b− a)Q3
k (5.61)
−[(4b− a)(12b− a)− C]Q2k − α2
) 1
2
where (k = 1, 2),
E = 2(4b− a)[(4b− a)b− C].
37
5.2.5 The Harmonic D-Extension
The fifth generalization is a mixture of the second and the third one
q1 + aq1 = −3q21 − 1
2q22,
q2 + bq2 = −q1q2 + Dq32
.(5.62)
In this case we have the Hamiltonians
H =p2
1
2+p2
1
2+q1q
22
2+ q3
1 +aq2
1
2+bq2
2
2+
D
2q22
, (5.63)
and
F = −2q2p1p2 + 2q1p22 −
1
4q42 − q2
1q22 − 2bq2
2q1 − (4b− a)(p22 + bq2
2) (5.64)
+2D
q22
q1 −D
q22
(4b− a),
the first Hamiltonian form
q1
q2
p1
p2
D
t
=
p1
p2
−3q21 − 1
2q22 − aq1
−q1q2 − bq2 + Dq32
0
=
=
0 0 1 0 0
0 0 0 1 0
−1 0 0 0 0
0 −1 0 0 0
0 0 0 0 0
︸ ︷︷ ︸
J0
3q21 + 1
2q22 + aq1
q1q2 + bq2 − Dq32
p1
p2
12q2
2
︸ ︷︷ ︸
∇H
, (5.65)
the second Hamiltonian form
q1
q2
p1
p2
D
t
=1
2q22
0 0 0 k l
0 0 k m n
0 −k 0 r s
−k −m −r 0 t
−l −n −s −t 0
︸ ︷︷ ︸
J1
38
×
2p22 − 2q1q
22 − 2bq2
2 + 2Dq22
−2p1p2 − q32 − 2q2
1q2 − 4bq2q1 − 2bAq2 − 4Dq1
q32
+ DAq32
−2q2p2
−2q2p1 + 4q1p2 − 2Ap2
2q1
q22
− Aq22
︸ ︷︷ ︸
∇F
, (5.66)
where
k = −q22 , l = 2q3
2p22, m = 2q1 − A, (5.67)
n = 2q32p1 − 4q1q
22p2 + 2Aq2
2p2, r = −p2, (5.68)
s = −2q1q42 + 2p2
2q22 + 2D − 2bq4
2, (5.69)
t = −2p1p2q22 − q5
2 − 2q21q
32 −
4Dq1q2
+2DA
q2− 4bq1q
32 + 2bAq3
2, (5.70)
with
A = (4b− a).
Canonically transformed (see Eqs.(5.45),(5.46),(5.47)) first Hamiltonian
H =1
2(Q1 −Q2)(P 2
1Q1 − P 22Q2) + [(4b− a)(12b− a)](Q1 +Q2)
4(5.71)
+(6b− a)(Q21 +Q2
2 +Q1Q2) + (Q1 +Q2)(Q21 +Q2
2)−D
8Q1Q2
+ 2b(4b− a
2)2,
and the second Hamiltonian
F =2
(Q2 −Q1)(P 2
2 − P 21 )Q1Q2 + 4Q1Q2(Q
21 +Q1Q2 +Q2
2) (5.72)
[(4b− a)(12b− a)]Q1Q2 + 4(6b− a)(Q1 +Q2)Q1Q2 −D(Q1 +Q2)
2Q1Q2,
determine the first couple of separated equations
∂W1k
∂Qk
=
√1
2Qk
(
4K1 − (4b− a)(12b− a)Q2k − 4(6b− a)Q3
k − 4Q4k (5.73)
+(4α1 − E)Qk −D
2Qk
]
) 1
2
,
and the second couple of separated equations
39
∂W2k
∂Qk
=
√1
2Qk
(
4K2Qk − 4Q4k − 4(6b− a)Q3
k (5.74)
−(4b− a)(12b− a)Q2k − α2 −
D
2Qk
]
) 1
2
,
correspondingly, where
E = 2b(4b− a)2, (k = 1, 2).
5.2.6 The Harmonic C-D Extension
As a sixth generalization we have a mixture of the first, the second and the
third generalizations considered above
q1 + aq1 = −3q21 − 1
2q22 + C,
q2 + bq2 = −q1q2 + Dq32
;(5.75)
Two Hamiltonians
H =p2
1
2+p2
1
2+q1q
22
2+ q3
1 +aq2
1
2+bq2
2
2+
D
2q22
− Cq1; (5.76)
and
F = −2q2p1p2 + 2q1p22 −
1
4q42 − q2
1q22 − 2bq2
2q1 − (4b− a)(p22 + bq2
2) (5.77)
+2D
q22
q1 −D
q22
(4b− a) + Cq22 ;
generate following equations; for the first Hamiltonian
q1
q2
p1
p2
C
D
t
=
p1
p2
−3q21 − 1
2q22 − aq1 + C
−q1q2 − bq2 + Dq32
0
0
=
40
=
0 0 1 0 0 0
0 0 0 1 0 0
−1 0 0 0 0 0
0 −1 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
︸ ︷︷ ︸
J0
3q21 + 1
2q22 + aq1 − C
q1q2 + bq2 − Dq32
p1
p2
−q11
2q22
︸ ︷︷ ︸
∇H
, (5.78)
and for the second one
q1
q2
p1
p2
C
D
t
=1
2q22
0 0 0 k 0 l
0 0 k m 0 n
0 −k 0 r 0 s
−k −m −r 0 0 t
0 0 0 0 0 0
−l −n −s −t 0 0
︸ ︷︷ ︸
J1
×
2p22 − 2q1q
22 − 2bq2
2 + 2Dq22
−2p1p2 − q32 − 2q2
1q2 − 4bq2q1 − 2bAq2 − 4Dq1
q32
+ DAq32
+ 2Cq2
−2q2p2
−2q2p1 + 4q1p2 − 2Ap2
q22
2q1
q22
− Aq22
︸ ︷︷ ︸
∇F
, (5.79)
where
k = −q22 , l = 2q3
2p22, m = 2q1 − A, (5.80)
n = 2q32p1 − 4q1q
22p2 + 2Aq2
2p2, r = −p2, (5.81)
s = −2q1q42 + 2p2
2q22 + 2D − 2bq4
2, (5.82)
t = −2p1p2q22 − q5
2 − 2q21q
32 −
4Dq1q2
+2DA
q2− 4bq1q
32 + 2bAq3
2 + 2Cq2, (5.83)
and
A = (4b− a).
41
By canonical transformation ((5.45),(5.46),(5.47)) we find expressions for Hamil-
tonian H,
H =1
2(Q1 −Q2)(P 2
1Q1 − P 22Q2) + [(4b− a)(12b− a)− 4C]
(Q1 +Q2)
4(5.84)
+(6b−a)(Q21+Q
22+Q1Q2)+(Q1+Q2)(Q
21+Q
22)−
D
8Q1Q2+[(
4b− a2
)[(4b−a)b−C]
and F,
F =2
(Q2 −Q1)(P 2
2 − P 21 )Q1Q2 + 4Q1Q2(Q
21 +Q1Q2 +Q2
2) (5.85)
+[(4b− a)(12b− a)− 4C]Q1Q2 + 4(6b− a)(Q1 +Q2)Q1Q2 −D(Q1 +Q2)
2Q1Q2
.
Corresponding Hamilton-Jacobi equations are separated in the form
∂W1k
∂Qk
=
√1
2Qk
(
4K1 − [(4b− a)(12b− a)− 4C]Q2k (5.86)
−4(6b− a)Q3k − 4Q4
k + (4α1 − E)Qk −D
2Qk
) 1
2
,
∂W2k
∂Qk
=
√1
2Qk
(
4K2Qk − 4Q4k − 4(6b− a)Q3
k (5.87)
−[(4b− a)(12b− a)− C]Q2k − α2 −
D
2Qk
) 1
2
,
where
E = 2b(4b− a)2, (k = 1, 2).
It is worth to note that in all above extensions, except Extension 3, the
arbitrary constants C,(and/or)D determining additional terms, appears in the
Hamiltonian formulation as independent canonical variables.
42
Chapter 6
RESONANCE SOLITONS IN AKNS HIERARCHY
In this section we construct one and two soliton solutions of the second and
third members of AKNS hierarchy with resonance soliton dynamics.
6.1 Hirota Bilinear Method in Soliton Theory
In 1971 Hirota introduced a new direct method for constructing soliton so-
lutions to integrable nonlinear evolution equations [13]. The idea is to make
transformation to new variables, so that in these variables a nonlinear evolution
equation become represented in the bilinear form , and multisoliton solutions
appear in particularly simple form. Multisoliton solutions can, of course de-
rived by many other methods, by the inverse scattering transform , dressing
method, Backlund and Darboux transformations, and so on. Particularly, the
Inverse Scattering Method(ISM) is very powerful, but at the same time it is
most complicated and needs information about analytic behaviour of scattering
data. Comparing with this, the advantage of Hirota’s method is its algebraic
rather than analytic structure [12], [74], [114]. It allows one to construct soliton
solution in a simple algebraic form avoiding analytic difficulties of ISM [73], [111],
[112], [113].
6.2 Bilinear Representation of Reaction-Diffusion System
The second member of the AKNS (6.1)is called the Reaction-Diffusion Equa-
tion
∂ye+ = ∂2
1e+ + λ
4e+e−e+,
−∂ye− = ∂2
1e− + λ
4e+e−e−.
(6.1)
For Hirota representation of this equation [20] we substitute
e± =
√
−8
λ
G±(x, t)
F (x, t), (6.2)
43
Then, partial derivatives of the ratio of two functions in the Hirota method [12]
can be represented in terms of so called Hirota derivatives (see Appendix I)
defined as
Dnx(f · g) = (∂x1
− ∂x2)nf(x1)g(x2)|x2=x1=x. (6.3)
In explicit form we have
Dx(f · g) = f ′g − g′f,D2
x(f · g) = f ′′g − 2f ′g′ + g′′f,
D3x(f · g) = f ′′′g − 3f ′′g′ + 3g′′f ′ − g′′′f,
...
(6.4)
Then we haved
dx(G±
F) =
Dx(G± · F )
F 2, (6.5)
d2
dx2(G±
F) = [
D2x(G
± · F )
F 2− G±
F
D2x(F · F )
F 2]. (6.6)
For bilinear representation of the system (6.1) we need the next derivatives
e±y =
√
−8
λ
Dy(G± · F )
F 2, (6.7)
e±x =
√
−8
λ
Dx(G± · F )
F 2, (6.8)
e±xx =
√
−8
λ[D2
x(G± · F )
F 2− G±
F
D2x(F · F )
F 2]. (6.9)
Substituting to the Reaction-Diffusion system (6.1) we can split the last one to
the couple of bilinear equations
(±Dy −D2x)(G
± · F ) = 0,
D2x(F · F ) = −2G+G−.
(6.10)
Then any solution of this system determines a solution of the Reaction-Diffusion
system (6.1). Simplest solution of bilinear system (6.10) has been derived in the
form [76]
G± = ±eη±
1 , F = 1 +e(η
+
1+η−
1)
(k+1 + k−1 )2
, (6.11)
where η±1 = k±1 x± (k±1 )2y + η±(0)1 . This solution determines soliton-like solution
of the Reaction-Diffusion system called the dissipaton [20], with exponentially
44
growing and decaying amplitudes. But for the product e+e− one have perfect
one-soliton shape
e+e− =8k2
λ cosh2[k(x− vy − x0)], (6.12)
of the amplitude k = (k+1 + k−1 )/2, propagating with velocity v = −(k+
1 −k−1 ), where the initial position x0 = − ln(k+
1 + k−1 )2 + η+(0)1 + η
−(0)1 . For two-
soliton(dissipaton) solution we have
G± = ±(eη±
1 + eη±
2 + α±1 e
η+
1+η−
1+η±
2 + α±2 e
η+
2+η−
2+η±
1 ), (6.13)
F = 1 +eη+
1+η−
1
(k+−11 )2
+eη+
1+η−
2
(k+−12 )2
+eη+
2+η−
1
(k+−21 )2
+eη+
2+η−
2
(k+−22 )2
+ βeη+
1+η−
1+η+
2+η−
2 , (6.14)
where
η±i = k±i x± (k±i )2y + η±i (0)
kabij = ka
i + kbj , (i, j = 1, 2), (a, b = +−)
α±1 =
(k±1 − k±2 )2
(k+−11 k
±∓21 )2
, α±2 =
(k±1 − k±2 )2
(k+−22 k
±∓12 )2
β =(k+
1 − k+2 )2(k−1 − k−2 )2
(k+−11 k
+−12 k
+−21 k
+−22 )2
.
6.3 Resonance Dynamics of Dissipatons
The degenerate case of this solution, when k+1 = k−1 ≡ p1, k
+2 = k−2 ≡ p2, can
be simplified in the form
e± = ±√
−8
λp+p−
p1 cosh θ2e±p2
1t + p2 cosh θ1e
±p22t
p2− cosh θ+ + p2
+ cosh θ− + 4p1p2 cosh(p+p−t), (6.15)
where p± ≡ p1 ± p2,θ± ≡ θ1 ± θ2, θi ≡ pi(x− x0i), (i = 1, 2). In this solution we
have substituted t instead of y variable. It allows us to interpret it dynamically in
terms of time evolution t = y. Then, reduced solution (6.15) describes a collision
of two dissipatons with identical amplitudes p+/2, moving in opposite directions
with equal velocities ‖v‖ = ‖p−‖, and creating the resonance bound state [76].
The lifetime of this state, T ≈ 2p2d/p+p−, linearly depends on the relative
distance d, where x01 = 0, x02 = d.
In a more general case, tractable analytically, when k±i > 0, (i = 1, 2), and
k+1 − k−1 > 0, k+
2 − k−2 > 0, k+1 − k−2 > 0, k+
2 − k−1 < 0, solution (6.13), (6.14)
45
describes collision of solitons with velocities v12 = −(k+1 − k−2 ) and v21 = −(k+
2 −k−1 ), correspondingly. Depending on the relative position’s shift, also in this
general case the resonance states can be created [76].
As a simplest example we consider conditions for decay of a dissipaton at
rest (v = 0) on two dissipatons with parameters (k1, v1) and (k2, v2). From the
conservation laws one obtains the following relations v21 = 4k2
2, v22 = 4k2
1 leading
to two possibilities :
(a)‖v1‖ = ‖v2‖. In this case ‖k1‖ = ‖k2‖, and both dissipatons have equal
masses M1 = M2 = M/2 and velocities, satisfying the critical values v2i =
4k2i , (i = 1, 2)
(b)‖v1‖ > ‖v2‖(without lose of generality). In this case v21 > 4k2
1 and v22 > 4k2
2
so that the initial dissipaton decays on a couple non-equal mass dissipatons.
The process of creation of resonant dissipaton is illustrated in Fig. 1.
-5 0 5 10 15 20-10
0
10
20
30
40
Figure 6.1: Resonant dissipaton creation
46
Figure 2 shows interaction of two dissipatons by exchange of a third dissipa-
ton.
20 30 40 50 60 70-60
-50
-40
-30
-20
-10
Figure 6.2: Exchange interactions of dissipatons
6.4 Geometrical Interpretation
Reaction-Diffusion system has a geometrical interpretation in a language of
constant curvature surfaces [76]. We define two-dimensional metric tensor in
terms of e+and e−,
gµν = eaµe
bνηab =
1
2(e+µ e
−ν + e+ν e
−µ ), (6.16)
where e±µ = e0µ ± e1µ = (e±0 , e±1 ), ηab = diag(−1, 1),
e±0 = ± ∂
∂xe±, e±1 ≡ e±, (6.17)
such that
g00 = −∂e+
∂x
∂e−
∂x, g11 = e+e−, g01 =
1
2(∂e+
∂xe− − e+∂e
−
∂x), (6.18)
implying identification x0 ≡ t(y), x1 ≡ x. It follows that when e± satisfy the
Reaction-Diffusion equations (6.1), this metric describes two-dimensional pseudo-
Riemannian space-time with constant curvature Λ:
R = gµνRµν = Λ. (6.19)
47
If we calculate the metric (6.16) for one soliton solution
ds2 =−8k2
λ cosh2 k(x− vt− x0)(6.20)
×[(k2 tanh2 k(x− vt)− 1
4v2)(dt)2 − (dx)2 − vdxdt]
then for ‖v‖ < 2‖k‖ ≡ ‖vmax‖, it shows a singularity (sign changing) at
tanh k(x− vt) = ± v
2k. (6.21)
It was shown [76], that this singularity (called the casual singularity )has phys-
ical interpretation in terms of black hole physics [106], [107] and relates with
resonance properties of solitons.
6.5 Dissipative solitons for the Third Flow
For the third flow of AKNS hierarchy we have the qubic dispersion system
∂te+ = ∂3
1e+ + 3λ
4e+e−∂1e
+,
∂te− = ∂3
1e− + 3λ
4e+e−∂1e
−,(6.22)
For the bilinear representation of this system first of all we represent functions
e(±)(x, t) satisfying Eqs.(6.22) in terms of three real functions G±, F . Hirota’s
derivatives are defined as before (6.3)(6.4). But for the third derivative term we
need also following expressions
∂3
∂x3(g
f) =
D3x(g · f)
f 2− 3[
D2x(g · f)
f 2
D2x(f · f)
f 2],
and
e±xxx =
√
−8
λ[D3
x(G± · F )
F 2− 3[
D2x(G
± · F )
F 2
D2x(F · F )
F 2]. (6.23)
Then, we have the bilinear form of Eqs.(6.22)
(Dt +D3x)(G
± · F ) = 0,
D2x(F · F ) = −2G+G−.
(6.24)
From the last equation we have for the product
U = e(+)e(−) =8
−λG+G−
F 2=
4
λ
D2x(F · F )
F 2=
8
λ
∂2
∂x2lnF. (6.25)
As in the canonical Hirota approach, we search solution of this bilinear system
in the form
48
G± = εG±1 + ε3G±
3 + ... , F = F0 + ε2F2 + ε4F4 + ... , (6.26)
where ε is a parameter which is not small. If we substitute G± and F to the
system 6.24 we get a sequence of equations in ε,
(Dt +D3x)(εG
±1 + ε3G±
3 + ...) · (F0 + ε2F2 + ε4F4 + ...) = 0,
D2x(F0 + ε2F2 + ε4F4 + ...) · (F0 + ε2F2 + ε4F4 + ...)
= −2(εG+1 + ε3G+
3 + ...)(εG−1 + ε3G−
3 + ...).
(6.27)
In the zero order approximation ε0 we have equation D2x(F0 · F0) = 0, with
an arbitrary constant solution F0 = constant. Without loss of generality we can
put this constant to the one. Indeed, the Hirota substitution is invariant under
multiplication of functions G and F with arbitrary function h(x,t):
e± =G±
F=hG±
hF=
h(εG±1 + ε3G±
3 + ...)
h(F0 + ε2F2 + ε4F4 + ...), (6.28)
so that we can always choose h = 1/F0.
1. For ε1 we have the system
(Dt +D3x)(G
±1 · F0) = 0, (6.29)
with solution
G±1 = ±eη±
1 , (6.30)
where
η±1 = k±1 x− (k±1 )3t+ η±(0)1 . (6.31)
2. Then ε2 equation
2D2x(1 · F2) = −2G+
1 G−1 , (6.32)
provides solution
F2 =e(η
+
1+η−
1)
(k+1 + k−1 )
2 . (6.33)
3. For ε3 the system is
(Dt +D3x)(G
±1 · F2 +G±
3 · 1) = 0 (6.34)
49
6.5.1 One Dissipative Soliton Solution
Simplest solution of this system G3 = 0 implies to take all higher order
terms Gn = 0, n = 3, 5, 7, ... and Fn = 0, n = 4, 6, 8, .... It is easy to check
that this truncated solution is an exact solution of our system,
G± = ±eη±
1 , F = 1 +e(η
+
1+η−
1)
k+1 + k−1
, (6.35)
where η±1 = k±1 x− (k±1 )3t+ η±(0)1 . It defines one dissipative soliton solution
of the system (6.22) in the form
e± = ±√
8
−Λ
|k±11|2
e±1
2(η+
1−η−
1)
coshη+
1+η−
1+φ11
2
, (6.36)
η+1 + η−1 = (k+
1 + k−1 )[x− vτ − x0], (6.37)
where
v = (k+1
2 − k+1 k
−1 + k−1
2), x0 =
η+1
(0)+ η+
1(0)
k+1 k
−1
,φ11
2= ln
1
k+−11
.
6.5.2 Two Dissipative Soliton Solution
Another, nontrivial choice for G3 we find as
G±3 = ±eη±
2 , (6.38)
where
η±2 = k±2 x− (k±2 )3t+ η±2 (0). (6.39)
4. In this case for ε4 we have the equation
2D2x(1 · F4) +D2
x(F2 · F2) = −2(G+1 G
−3 +G+
3 G−1 ), (6.40)
with following solution
F4 =e(η
+
1+η−
2)
(k+1 + k−1 )2
+e(η
+
2+η−
1)
(k+2 + k−1 )2
. (6.41)
5. For ε5 for the system
(Dt +D3x)(G
±5 · 1 +G±
3 · F2 +G±1 · F4) = 0 (6.42)
50
and we find solution
G±5 = ±α1
±eη+
1+η−
1+η±
2 , (6.43)
where coefficients
α±1 =
(k±1 − k±2 )2
(k+−11 k
±∓21 )2
. (6.44)
6. At the next level ε6 equation
2D2x(1 · F6) + 2D2
x(F2 · F4) = −2(G+1 G
−5 +G+
3 G−3 +G+
5 G−1 ), (6.45)
admits solution
F6 =e(η
+
2+η−
2)
(k+2 + k−2 )
2 . (6.46)
7. For ε7 it is
(Dt +D3x)(G
±7 · 1 +G±
5 · F2 +G±3 · F4 +G±
1 · F6) = 0, (6.47)
and we find
G±7 = ±α2
±e(η+
2+η−
2+η±
1), (6.48)
where
α±2 =
(k±1 − k±2 )2
(k+−22 k
±∓12 )2
. (6.49)
8. Next level ε8 gives equation
2D2x(1 · F8) + 2D2
x(F2 · F6) +D2x(F4 · F4) (6.50)
= −2(G+1 G
−7 +G+
3 G−5 +G+
5 G−3 +G+
7 G−1 )
with solution
F8 = βe(η+
1+η−
1+η+
2+η−
2), (6.51)
where
β =(k+
1 − k+2 )2(k−1 − k−2 )2
(k+−11 k
+−12 k
+−21 k
+−22 )2
. (6.52)
9. For higher order terms in ε we can choose Gn = 0, with index n =
9, 11, 13, ... and Fn = 0, with index n = 10, 12, 14, .... Then, by direct
substitution we checked that with this choice, bilinear equations are satis-
fied in all orders of ε. Therefore we have an exact solution.
51
This solution is the two soliton solution in the form
G± = ±(eη±
1 + eη±
2 + α±1 e
η+
1+η−
1+η±
2 + α±2 e
η+
2+η−
2+η±
1 ), (6.53)
F = 1 +eη+
1+η−
1
(k+−11 )2
+eη+
1+η−
2
(k+−12 )2
+eη+
2+η−
1
(k+−21 )2
+eη+
2+η−
2
(k+−22 )2
+ βeη+
1+η−
1+η+
2+η−
2 , (6.54)
where
η±i = k±i x− (k±i )3t+ η±(0)i ,
kabij = ka
i + kbj , (i, j = 1, 2), (a, b = +−),
α±1 =
(k±1 − k±2 )2
(k+−11 k
±∓21 )2
, α±2 =
(k±1 − k±2 )2
(k+−22 k
±∓12 )2
, (6.55)
β =(k+
1 − k+2 )2(k−1 − k−2 )2
(k+−11 k
+−12 k
+−21 k
+−22 )2
. (6.56)
6.5.3 The MKdV Reduction
From Section 5.3,we know that the third order of AKNS flow under special
reduction
e+ = e− = U (6.57)
reduces to MKdV equation (3.34). For the bilinear equation (6.24), it means the
following reduction
G+ ≡ G− ≡ G, (6.58)
under this reduction
(Dt +D3x)(G · F ) = 0
D2x(F · F ) = −2G2
(6.59)
we have
eη+
1 = eη−
1 ≡ eη, (6.60)
k+1 = k−1 ≡ k, η1
+(0) = η1−(0), (6.61)
and for one-soliton solution of MKdV
e+ = e− = U =
√
8
−Λ|k| 1
cosh(η + φ11
2), (6.62)
or
U(x, t) =
√
8
−Λ
|k|cosh k(x− k2τ − x0)
, (6.63)
52
where
x0 = −η(0) + φ11
2
k,φ11
2= ln
1
k+−11
.
For two soliton solution we find the reduction
η+(0)1 = η
−(0)1 = η
(0)1 ,
η+(0)2 = η
−(0)2 = η
(0)2 ,
k+1 = k−1 ≡ k1,
k+2 = k−2 ≡ k2,
(6.64)
and
G = (eη1 + eη2 + α1e2η1+η2 + α2e
2η2+η1), (6.65)
F = 1 +eη1
4k12 + 2
eη1+η2
(k1 + k2)2 +
e2η2
4k12 + βe2η1+2η2 , (6.66)
where
α1 =(k1 − k2)
2
4k12(k1 + k2)
2 , α2 =(k1 − k2)
2
4k22(k1 + k2)
2 ,
β =(k1 − k2)
4
16k12k2
2(k1 + k2)4.
It gives 2-soliton solution of MKdV equation in the form
U =
√
8
−Λ
G
F=
√
8
−Λ
(eη1 + eη2 + α1e2η1+η2 + α2e
2η2+η1)
1 + eη1
4k12 + 2 eη1+η2
(k1+k2)2 + e2η2
4k12 + βe2η1+2η2
(6.67)
or
U =
√
8
−Λ
2k1k2|k21 − k2
2|[k2 cosh η1 + k1 cosh η2]
(k1 − k2)2 cosh(η1 + η2) + (k1 + k2)
2 cosh(η1 − η2) + 4k1k2
(6.68)
where
η1 = η1 + ψ + φ1 = k1(x− k21τ −X1
0 ), X10 = 1
k1η1
(0) + ln |k1−k2||k1+k2| −
12ln 4k2
1,
η2 = η2 + ψ + φ2 = k2(x− k22τ −X2
0 ), X20 = 1
k2η2
(0) + ln |k1−k2||k1+k2| −
12ln 4k2
2.
6.5.4 The MKdV-KdV Mixed Reduction
As it was shown in Section 5.3, the third order of AKNS flow under the special
reduction (3.38) gives the mixed KdV-MKdV equation (3.39)
∂t2U = ∂3xU +
3λ
4(α + β)(αU2∂xU + β U∂xU). (6.69)
53
To produce bilinear representation for such mixed equation, this reduction
can be imposed on the bilinear equations (6.24)
(Dt +D3x)(G
± · F ) = 0,
D2x(F · F ) = −2G+G−,
(6.70)
in the form
e+ = (α+ β)U = G+
F,
e− = αU + β = G−
F,
(6.71)
or
G+ = (α + β)G,
G− = αG+ βF.(6.72)
Under this reduction we get bilinear representation for Eq.(6.69) as follows
(Dt +D3x)(G · F ) = 0,
12D2
x(F · F ) = (α + β)αG2 + (α + β)βGF.(6.73)
Solution of this system provides solution of Eq(6.69) according formula
αU2 + βU =1
2(α + β)
D2x(F · F )
F 2=
1
(α+ β)(lnF )xx. (6.74)
Then, for one-soliton solution we have
U =G
F=
eη1
1 + (α+β)βk21
eη1 + 14k2
1
[(α + β)β + (α+β)2β2
k21
]e2η1
, (6.75)
or
U =e−φ
2 cosh(η1 + φ) + (α+β)β
k21
eφ, (6.76)
where
φ =1
2ln
1
4k21[(α + β)β + (α+β)2β2
k21
]. (6.77)
54
Chapter 7
THE KADOMTSEV-PETVIASHVILI MODEL
7.1 2+1 Dimensional Reduction of AKNS
AKNS hierarchy allows us to develop also a new method to find solution
for (2+1) and higher dimensional integrable systems, namely the Kadomtsev-
Petviashvili (KP)equation. Kadomtsev-Petviashvili equation is one of a few soli-
ton equations which describes physical phenomena in two-dimensional space [27].
The equation was presented by Kadomtsev and Petviashvili to discuss the sta-
bility of one-dimensional soliton in a nonlinear media with weak dispersion. It
has been explored recently in plasma physics, hydrodynamics, string theory and
low-dimensional gravity [120], [121], [122], [123], [124]. The hierarchy of KP
equations [104], [131] has a reach mathematical structure related with complex
analysis and Riemann surfaces, pseudo-differential operators and algebraic ge-
ometry [105], [125], [127], [128], [28].
Depending on sign of dispersion, two types of the KP equations are known.
The minus sign in the right side of the KP corresponds to the case of negative
dispersion and called KPII. Now we describe a new relation of KPII with AKNS
hierarchy [117] discussed in Chapter 6 and construct corresponding solutions.
Let us consider the pair of functions e+(x, y, t), e−(x, y, t) satisfies the following
second (7.1) and third (7.2) equations of the hierarchy.
e+y = e+xx + λ4e+e−e+,
−e−y = e−xx + λ4e+e−e−,
(7.1)
e+t = e+xxx + 3λ4e+e−e+x ,
e−t = e−xxx + 3λ4e+e−e−x .
(7.2)
Differentiating according to t and y, Eqs. (7.1) and (7.2) correspondingly, we can
see that they are compatible.
55
Theorem 7.1.0.1 Let the functions e+(x, y, t) and e−(x, y, t) are simultaneouslysolutions of the equations (7.1) and (7.2). Then the function U(x, y, t) = e+e−
satisfies the Kadomtsev-Petviashvili (KPII) equation
(4Ut +3λ
4(U2)x + Uxxx)x = −3Uyy. (7.3)
Proof: We take the derivative of U according to y variable
Uy = e+y e− + e+e−y (7.4)
and substituting e+y and e−y from the system (7.1) we have
Uy = (e+x e− − e−x e+)x, (7.5)
Uyy = (e+xxxe− + e−xxxe
+ − (e+x e−x )x) +
λ
2UxU. (7.6)
In a similar way Ut is
Ut = e+t e− + e+e−t (7.7)
and after substitution of e+t and e−t we get
Ut = −(e+xxxe− +
3λ
4Ue−e−x + e−xxxe
+ +3λ
4Ue−x e
+), (7.8)
and
Uxt = −(e+xxxe− + e−xxxe
+ +3λ
4UUx)x. (7.9)
Combining above formulas together
4Uxt + 3Uyy = [−e+xxxe− − e−xxxe
+ − 3λ
2UUx − 3(e+x e
−x )x]x, (7.10)
and using
Uxxx = e+xxxe− + e−xxxe
+ + 3e+xxe−x + 3e+x e
−xx, (7.11)
we get KPII (7.3)
4Uxt + 3Uyy = −3λ
4(U2)xx − Uxxxx. (7.12)
Like the KdV equation, KPII is an infinite dimensional Hamiltonian system
admitting (2+1) dimensional soliton solution [127], [128], [28]. Each soliton is
a planar wave similar to (1+1) KdV type soliton, but traveling in an arbitrary
direction in the x-y plane. We can use the above Theorem to generate solutions
of KPII in terms of solutions of equations (7.1) and (7.2).
56
7.2 Bilinear Representation of KPII and AKNS flows
Using bilinear representations for systems (7.1) and (7.2) and Theorem 7.1.0.1
we can find bilinear representation for KPII. The Reaction-Diffusion system (7.1)
can be represented in Hirota bilinear form as
(±Dy −D2x)(G
± · F ) = 0,
D2x(F · F ) = −2G+G−.
(7.13)
In a similar way the system of the third flow equations (6.22) has the following
bilinear form
(Dt +D3x)(G
± · F ) = 0,
D2x(F · F ) = −2G+G−.
(7.14)
Any solution G±(x, y), F (x, y) of bilinear equations (7.13) satisfies the system of
equations (7.1) for e(±)(x, y), while any solution G±(x, t), F (x, t), of Eqs.(7.14)
satisfies the system (6.22) for e(±)(x, t). Now we consider G± and F as functions
of three variables G(±) = G(±)(x, y, t), F = F (x, y, t), and require for these
functions to be solution of both bilinear systems (7.13), (7.14) simultaneously.
Since the second equation in both systems (7.13),(7.14)is the same, it is sufficient
to consider the next bilinear system [135]
(±Dy −D2x)(G
± · F ) = 0,
(Dt +D3x)(G
± · F ) = 0,
D2x(F · F ) = −2G+G−.
(7.15)
Then, according to Theorem 7.1.0.1, any solution of this system generates solu-
tion of KPII.
From the last equation we can derive expression for solution of KPII (7.3),
directly in terms of function F only
U = e(+)e(−) =8
−λG+G−
F 2=
4
λ
D2x(F · F )
F 2=
8
λ
∂2
∂x2lnF (7.16)
As in the canonical Hirota approach we search solution of this bilinear system
in the form
G± = εG±1 + ε3G±
3 + ... , F = F0 + εF2 + ε4F4 + ... , (7.17)
where ε is a parameter which is not small. If we substitute this expansion to
system (7.15) we get a sequence of equations
57
(±Dy −D2x)(εG
±1 + ε3G±
3 + ...).(F0 + ε2F2 + ε4F4 + ...) = 0,
(Dt +D3x)(εG
±1 + ε3G±
3 + ...).(F0 + ε2F2 + ε4F4 + ...) = 0,
D2x(1 + ε2F2 + ε4F4 + ...).(F0 + ε2F2 + ε4F4 + ...)
= −2(εG+1 + ε3G+
3 + ...)(εG−1 + ε3G−
3 + ...).
(7.18)
In the zero order level we have equation D2x(F0.F0) = 0 with an arbitrary con-
stant solution F0 = const. Like before, without lose of generality, we can put this
constant to one, since the Hirota substitution is invariant under multiplication
of functions G and F with an arbitrary function h(x,t):
U =G±
F=hG±
hF=
h(εG±1 + ε3G±
3 + ...)
h(F0 + ε2F2 + ε4F4 + ...)(7.19)
which we choose as h = 1/F0.
1. For ε1 we have the system
(±Dy −D2x)(G
±1 · F0) = 0,
(Dt +D3x)(G
±1 · F0) = 0,
(7.20)
with solution
G±1 = ±eη±
1 , (7.21)
where
η±1 = k±1 x± (k±1 )2y − (k±1 )3t+ η±(0)1 (7.22)
2. For ε2 equation
2D2x(1 · F2) = G+
1 G−1 (7.23)
after integration
F2 =e(η
+
1+η−
1)
(k+1 + k−1 )
2 . (7.24)
3. For ε3 the system is
(±Dy −D2x)(G
±1 · F2 +G±
3 · 1) = 0,
(Dt +D3x)(G
±1 · F2 +G±
3 · 1) = 0.(7.25)
58
Simplest solution of this system G3 = 0, implies to take all higher order
terms Gn = 0, n = 3, 5, 7, ... and Fn = 0, n = 4, 6, 8, .... It is easy to check
that this truncated solution is an exact solution of our system,
G± = ±eη±
1 , F = 1 +e(η
+
1+η−
1)
k+1 + k−1
, (7.26)
where η±1 = k±1 x± (k±1 )2y − (k±1 )3t+ η±(0)1 , defining one-soliton solution of
KPII according to Eq.(7.16)
U =2(k+
1 + k−1 )2
λ cosh2 12[(k+
1 + k−1 )x+ (k+1
2 − k−12)y − (k+
13+ k−1
3)t+ γ]
, (7.27)
where γ = − ln(k+1 + k−1 )2 + η
+(0)1 + η
−(0)1
Another, nontrivial choice for G3 we find as
G±3 = ±eη±
2 , (7.28)
where
η±2 = k±2 x± (k±2 )2y − (k±2 )3t+ η±(0)2 (7.29)
4. In this case for ε4 we have the equation
2D2x(1 · F4) +D2
x(F2 · F2) = −2(G+1 G
−3 +G+
3 G−1 ), (7.30)
and solution
F4 =e(η
+
1+η−
2)
(k+1 + k−1 )2
+e(η
+
2+η−
1)
(k+2 + k−1 )2
(7.31)
5. For ε5 it gives
(±Dy −D2x)(G
±5 · 1 +G±
3 · F2 +G±1 · F4) = 0,
(Dt +D3x)(G
±5 · 1 +G±
3 · F2 +G±1 · F4) = 0,
(7.32)
and we find
G±5 = ±α1
±eη+
1+η−
1+η±
2 , (7.33)
where
α±1 =
(k±1 − k±2 )2
(k+−11 k
±∓21 )2
. (7.34)
59
6. At the next level ε6 a single equation is
2D2x(1 · F6) + 2D2
x(F2 · F4) = −2(G+1 G
−5 +G+
3 G−3 +G+
5 G−1 ) (7.35)
and solution
F6 =e(η
+
2+η−
2)
(k+2 + k−2 )
2 (7.36)
7. For ε7 it is
(±Dy −D2x)(G
±7 · 1 +G±
5 · F2 +G±3 · F4 +G±
1 · F6) = 0,
(Dt +D3x)(G
±7 · 1 +G±
5 · F2 +G±3 · F4 +G±
1 · F6) = 0,(7.37)
and we find
G±7 = ±α2
±e(η+
2+η−
2+η±
1), (7.38)
where
α±2 =
(k±1 − k±2 )2
(k+−22 k
±∓12 )2
. (7.39)
8. Next level ε8 gives
2D2x(1 · F8) + 2D2
x(F2 · F6) +D2x(F4 · F4) (7.40)
= −2(G+1 G
−7 +G+
3 G−5 +G+
5 G−3 +G+
7 G−1 ),
and
F8 = βe(η+
1+η−
1+η+
2+η−
2), (7.41)
where
β =(k+
1 − k+2 )2(k−1 − k−2 )2
(k+−11 k
+−12 k
+−21 k
+−22 )2
. (7.42)
9. For higher order in ε we can choose Gn = 0, with index n = 9, 11, 13, ..., and
Fn = 0, with index n = 10, 12, 14, .... By direct substitution we checked that
with this choice bilinear equations are satisfied in all orders of ε. Therefore
we have an exact solution.
7.3 Two Soliton Solutions
This solution is two soliton solution in the form [135]
G± = ±(eη±
1 + eη±
2 + α±1 e
η+
1+η−
1+η±
2 + α±2 e
η+
2+η−
2+η±
1 ), (7.43)
60
F = 1 +eη+
1+η−
1
(k+−11 )2
+eη+
1+η−
2
(k+−12 )2
+eη+
2+η−
1
(k+−21 )2
+eη+
2+η−
2
(k+−22 )2
+ βeη+
1+η−
1+η+
2+η−
2 , (7.44)
where
η±i = k±i x± (k±i )2y − (k±i )3t+ η±(0)i ,
kabij = ka
i + kbj , (i, j = 1, 2), (a, b = +−),
α±1 =
(k±1 − k±2 )2
(k+−11 k
±∓21 )2
, α±2 =
(k±1 − k±2 )2
(k+−22 k
±∓12 )2
,
β =(k+
1 − k+2 )2(k−1 − k−2 )2
(k+−11 k
+−12 k
+−21 k
+−22 )2
.
According to Eq.(7.16) it provides two-soliton solution of KPII
U = e+e− =−8
Λ
∂2
∂x2lnF. (7.45)
7.4 Degenerate Four-Soliton Solution
For KPII another bilinear form, only in terms of one function F, is known [74]
(DxDt +D4x ±D2
y)(F · F ) = 0 (7.46)
where
U = 2∂2
∂x2lnF. (7.47)
Thus it is natural to compare soliton solutions of our bilinear equations (7.15)
with the ones given by above bilinear equation [135]. To solve equation (7.46)we
consider
F = 1 + εF1 + ε2F2 + ... . (7.48)
1. For ε1 we have the equation
(DxDt +D4x +D2
y)(1 · F1) = 0, (7.49)
with the solution
F1 = eη1 , (7.50)
where
η1 = k1x+ Ω1y − ω1t+ η01, (7.51)
and dispersion
k1ω1 + k41 + Ω2
1 = 0. (7.52)
61
2. For ε2 we have equation
(DxDt +D4x +D2
y)(1 · 2F2 + F1 · F1) = 0. (7.53)
The simplest solution of this equation is the trivial one
F2 = 0. (7.54)
Then, truncating the series by putting all higher order terms in ε to zero,
Fn = 0, (n = 2,3,...), we have one soliton solution of KPII (7.3):
U =k1
2 cosh2 12[(k1x+ (k+
12 − k−1
2)y − (k+
13+ k−1
3)t+ γ]
(7.55)
If we consider one soliton solution of equation (7.46), we realize that it
coincides with our one soliton solution (7.27)
U =2(k+
1 + k−1 )2
λ cosh2 12[(k+
1 + k−1 )x+ (k+1
2 − k−12)y − (k+
13+ k−1
3)t+ γ]
(7.56)
where γ = − ln(k+1 + k−1 )2 + η
+(0)1 + η
−(0)1 . But two soliton solution of
equation (7.46)doesn’t correspond to our two-soliton solution (7.43),(7.44).
Appearance of four different terms eη±
i +η±
k in equations (7.43),(7.44), sug-
gest that our two-soliton solution should correspond to some degenerate
case of four soliton solution of Eq(7.46)(We thank Prof. J. Hietarinta for
this suggestion).
To construct two soliton solution we choose another solution of bilinear
equation (7.46)
F2 = eη2 (7.57)
where
η2 = k2x+ Ω2y − ω2t+ η02 (7.58)
3. For ε3 we have the equation
(DxDt +D4x +D2
y)(1 · 2F3 + 2F1 · F2) = 0, (7.59)
with the solution
F3 = α12eη1+η2 , (7.60)
where
α12 = −(k1 − k2)(ω1 − ω2) + (k1 − k2)4 + (Ω1 − Ω2)
2
(k1 + k2)(ω1 + ω2) + (k1 + k2)4 + (Ω1 + Ω2)
2 . (7.61)
62
4. For ε4 equation
(DxDt +D4x +D2
y)(1 · 2F4 + F1 · 2F3 + F2 · F2) = 0 (7.62)
has solution
F4 = eη3 , (7.63)
where
η3 = k3x+ Ω3y − ω3t+ η03. (7.64)
5. For ε5 equation
(DxDt +D4x +D2
y)(1 · 2F5 + F1 · 2F4 + F2 · 2F3) = 0 (7.65)
admits
F5 = α13eη1+η3 , (7.66)
where
α13 = −(k1 − k3)(ω1 − ω3) + (k1 − k3)4 + (Ω1 − Ω3)
2
(k1 + k3)(ω1 + ω3) + (k1 + k3)4 + (Ω1 + Ω3)
2 . (7.67)
6. For ε6 equation is
(DxDt +D4x +D2
y)(1 · 2F6 + F1 · 2F5 + F2 · 2F4 + F3 · F3) = 0, (7.68)
with solution
F6 = α23eη1+η3 , (7.69)
where
α23 = −(k2 − k3)(ω2 − ω3) + (k2 − k3)4 + (Ω2 − Ω3)
2
(k2 + k3)(ω2 + ω3) + (k2 + k3)4 + (Ω2 + Ω3)
2 . (7.70)
Now we will do the special parameterizations of our solution
k1 = k+1 + k−1 , ω1 = −4(k+3
1 + k−31 ), Ω1 =
√3(k+2
1 − k−21 ),
k2 = k+2 + k−2 , ω2 = −4(k+3
2 + k−32 ), Ω2 =
√3(k+2
2 − k−22 ),
k3 = k+1 + k−2 , ω3 = −4(k+3
1 + k−32 ), Ω3 =
√3(k+2
1 − k−22 )
k4 = k+2 + k−1 , ω4 = −4(k+3
2 + k−31 ), Ω4 =
√3(k+2
2 + k−21 ),
(7.71)
63
Then, substituting these parameterizations we find that
α13 = 0 ⇒ F5 = 0 (7.72)
α23 = 0 ⇒ F6 = 0 (7.73)
7. For ε7 we have the equation
(DxDt +D4x +D2
y)(1 · 2F7 + F1 · 2F6 + F2 · 2F5 + F3 · 2F4) = 0, (7.74)
with the solution
F7 = eη4 , (7.75)
where
η4 = k4x+ Ω4y − ω4t+ η04. (7.76)
8. For ε8 the system
(DxDt +D4x+D2
y)(1 ·2F8+F1 ·2F7+F2 ·2F6+F3 ·2F5+F4 ·F4) = 0, (7.77)
gives solution
F8 = α14eη1+η4 , (7.78)
where
α14 = −(k1 − k4)(ω1 − ω4) + (k1 − k4)4 + (Ω1 − Ω4)
2
(k1 + k4)(ω1 + ω4) + (k1 + k4)4 + (Ω1 + Ω4)
2 . (7.79)
After the parameterizations given above, we also get
α14 = 0 ⇒ F8 = 0 (7.80)
9. For ε9 we have equation
(DxDt+D4x+D2
y)(1·2F9+F1 ·2F8+F2 ·2F7+F3 ·2F6+F4 ·2F5) = 0, (7.81)
64
with solution
F8 = α24eη2+η4 , (7.82)
where
α24 = −(k2 − k4)(ω2 − ω4) + (k2 − k4)4 + (Ω2 − Ω4)
2
(k2 + k4)(ω2 + ω4) + (k2 + k4)4 + (Ω2 + Ω4)
2 , (7.83)
which also leads to
α24 = 0 ⇒ F8 = 0. (7.84)
10. For ε10 equation
(DxDt +D4x +D2
y)(1 · 2F10 + F1 · F9 + F2 · 2F8 (7.85)
+F3 · 2F7 + F4 · 2F6 + F5 · F5) = 0,
is satisfied by solution
F10 = 0. (7.86)
11. For ε11 we have the equation
(DxDt +D4x +D2
y)(1 · 2F11 + F1 · 2F10 + F2 · 2F9 (7.87)
+F3 · 2F8 + F4 · 2F7 + F4 · 2F7) = 0,
and corresponding solution
F11 = α34eη3+η4 , (7.88)
α34 = −(k3 − k4)(ω3 − ω4) + (k3 − k4)4 + (Ω3 − Ω4)
2
(k3 + k4)(ω3 + ω4) + (k3 + k4)4 + (Ω3 + Ω4)
2 . (7.89)
When it is checked for higher order terms we find that
F12 = F13 = ... = 0. (7.90)
Thus, we have degenerate four-soliton solution of equations ( 7.46 )
F = 1 + eη1 + eη2 + eη3 + eη4 + α12eη1+η2 + α34e
η3+η4 . (7.91)
The above consideration shows that our two-soliton solution of KP-II corresponds
to the degenerate four soliton solution in the canonical Hirota form (7.46). More-
over, it allows us to find new four virtual soliton resonance for KPII.
65
7.5 Resonance Interaction of Planar Solitons
In section 7.3 we constructed two soliton solution of the KPII equation. Choos-
ing different values of parameters we find resonance character of our soliton inter-
action. For the next choice of parameters k+1 = 2, k−1 = 1, k+
2 = 1, k−2 = 0.3, and
vanishing value of the position shift constants, we obtained two soliton solution
moving in the plane with constant velocity, with creation of the four, so called
virtual solitons (solitons without asymptotic states at infinity [115]).
-20 -10 0 10 20-20
-10
0
10
20
Figure 7.1: Resonance KP soliton dynamics
Fig. 7.1-7.5 illustrates that at the negative time, the four virtual solitons are
decreasing in the size and at time zero collapse and disappear completely. Then,
when time positively growing, they start to grow in the size.
66
-20 -10 0 10 20-20
-10
0
10
20
Figure 7.2: Resonance KP soliton dynamics
-20 -10 0 10 20-20
-10
0
10
20
Figure 7.3: Resonance KP soliton dynamics
67
-20 -10 0 10 20-20
-10
0
10
20
Figure 7.4: Resonance KP soliton dynamics
-20 -10 0 10 20-20
-10
0
10
20
Figure 7.5: Resonance KP soliton dynamics
68
-15 -10 -5 0 5 10 15-10
-5
0
5
10
Figure 7.6: One Virtual Soliton Solution
On Fig 7.6 we show two soliton solution with parameters k+1 = 2, k−1 = 1, k+
2 =
0.001, k−2 = 0.001 which includes only one virtual soliton. This virtual soliton
can be considered as created from the pair of real solitons and is decaying into a
pair of real solitons.
The resonance character of our planar soliton interactions is related with
resonance nature of dissipatons considered in Chapter 6, section 6.3. It has
been reported also in several systems like the Sawada-Kotera equation [118] ,
the Boussinesq [116] type equation, the KP equation [75]. But the four virtual
soliton resonance does not seem to have been done for KPII [134] prior our work
[135]. Experimentally, the soliton resonance of ion-acoustic solitons [119] has
been observed.
In Fig. 7.7 we show two solitons observed in the ocean with similar to the
KP structure.
69
-20 -10 0 10 20-20
-10
0
10
20
Figure 7.7: Solitons in the ocean
70
Chapter 8
CONCLUSIONS
In the last decades, it has been a very exciting time of developing the soliton
theory, and quickly expanding wide field of applications to nonlinear phenomena
of this abstract concept of integrable Hamiltonian systems with a specific struc-
ture of the phase space. Now it is difficult to find the subject of natural sciences
where solitons and other non-perturbative approaches have not been applied yet.
From microscale of the elementary particle physics and quantum theory until
macroscale of the cosmology, solitons create a new paradigma of the nonlinear
world, similar to the role of the harmonic oscillator in the linear world.
In the present thesis we considered some aspects of integrability in finite di-
mensional Hamiltonian systems, their relations with soliton equations and new
type of resonance soliton dynamics. We found new hierarchy, mixing two famous
soliton equations as the KdV and MKdV equations, corresponding recursion op-
erator and soliton solution. The finite dimensional reduction of soliton equations
for the stationary flows in the form of the Henon-Heiles system we extended with
several terms and found corresponding separation of variables in the Hamilton-
Jacobi formalism and the bi-Hamiltonian structure.
We constructed the Hirota bilinear representation for some systems of soliton
equations with third order dispersion and one and two soliton solutions. We
applied these bilinear representations to integrate 2+1 dimensional KdV model
known as KPII, and found new resonance character of its soliton interactions.
We hope that finite dimensional models considered in this thesis due to the
exact solvability can help one to understand better the nature of transition from
integrability to the chaos. The idea to use couple of equations from the AKNS
hierarchy, which we explored in our study, can be applied also to multidimensional
sytems with zero curvature structure. These type of systems are known as non-
Maxwell gauge theories, or the Chern-Simons theories and have applications in
many areas of physics and mathematics.
71
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APPENDIX A
HIROTA DERIVATIVES AND ITS PROPERTIES
In this Appendix we list some properties of the Hirota derivative operators Dt,Dx
defined by equation
Dnx(f · g) = (∂x1
− ∂x2)nf(x1)g(x2)|x2=x1=x (A.1)
or in more general form
Dnt D
mx (f.g) =
(∂
∂t− ∂
∂t′
)n (∂
∂x− ∂
∂x′
)n
f(t, x)g(t′, x′)|t=t′,x=x′. (A.2)
From the definition above we can find the general expression for n-th Hirota
derivative
Dnx(f.g) = (∂x1
− ∂x2)nf(x1)g(x2)|x2=x1=x =
n∑
k=0
n
k
∂kx1∂(n−k)
x2f(x)g(x),
(A.3)
or
Dnx(f.g) =
n∑
k=0
n
k
f (n−k)(x)g(k)(x)(−1)k. (A.4)
For the first few derivatives we have explicitly
Dx(f.g) = f ′g − g′f ,D2
x(f.g) = f ′′g − 2f ′g′ + g′′f ,
D3x(f.g) = f ′′′g − 3f ′′g′ + 3g′′f ′ − g′′′f ,
D4x(f.g) = f (IV )g − 4f ′′′g′ + 6f ′′g′′ − 4f ′g′′′ + fg(IV ) ,
...
(A.5)
The following properties are easily seen from the definition
1. Dmx (f.1) = ( ∂
∂x)mf
83
2. Dmx (f.g) = (−1)mDm
x (g.f)
3. Dmx (f.f) = 0 for odd m.
4. D2x(f.f) = 2f ′′f − 2f ′2
5. Dmx (f.g) = Dm−1
x (fx.g − f.gx)
6. DxDt(f.f) = 2Dx(ft.f) = 2Dt(fx.f) for even m.
7. Dmx (ep1x.ep2x) = (p1 − p2)
me(p1+p2)x
8. Dmx (eΩ1t+p1x.eΩ2t+p2x) = (p1 − p2)
me(Ω1+Ω2)t+(p1+p2)x
9. Dnt (eΩ1t+p1x.eΩ2t+p2x) = (Ω1 − Ω2)
ne(Ω1+Ω2)t+(p1+p2)x
10. Let P (Dt, Dx) be a polynomial of Dt and Dx, we have
P (Dt, Dx)(eΩ1t+p1x.eΩ2t+p2x) = P (Ω1 − Ω2, p1 − p2)e
(Ω1+Ω2)t+(p1+p2)x
11. e(εDx)(f(x).g(x)) =∑∞
k=0εk
k!Dn
x(f.g) =∑∞
k=0εk
k!
∑kn=0
n
k
(−1)k−nf (n)(x)g(k−n)(x)
=
∞∑
n=0
f (n)(x)
n!εn
∞∑
m=0
g(m)(x)
m!(−ε)m
where k = m+n. As a result we get that this is equal to
= f(x+ ε)g(x− ε)
12. Dx(fg.h) = (∂f
∂x)gh+ fDx(g.h)
13. D2x(fg.h) = (∂2f
∂x2 )gh+ 2(∂f
∂x)Dx(g.h) + fD2
x(g.h)
14. Dmx ((epxf).(epxg)) = e2pxDm
x (f.g)
The following formulas are useful for transforming nonlinear differential
equations into bilinear forms.
15. ∂∂x
( g
f) = Dx(f.g)
f2
16. ∂2
∂x2 (g
f) = D2
x(g.f)f2 − g
f
D2x(g.f)f2
17. ∂3
∂x3 (g
f) = D3
x(g.f)f2 − 3[D2
x(g.f)f2
D2x(f.f)f2 ]
18. ∂2
∂x2 (logf) = D2x(f.f)2f2
19. ∂4
∂x4 (logf) = D4x(f.f)2f2 − 6[D2
x(f.f)2f2 ]2
84